ON CLUSTER ALGEBRAS WITH COEFFICIENTS AND 2-CALABI-YAU CATEGORIES

CHANGJIAN FU AND BERNHARD KELLER

Abstract. Building on work by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in ‘Cluster algebras IV: Coefficients’. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold.

1. Introduction

In this article, we pursue the representation-theoretic approach to Fomin-Zelevinsky’s cluster algebras [19] [20] [8] [21] developed by Marsh-Reineke-Zelevinsky [35], Buan-Marsh-Reineke-Reiten-Todorov [6] [7], Caldero-Chapoton and Caldero-Keller [9] [10] [11], Geiss-Leclerc-Schröer [25] [26] [23], [2] and many others, cf. the surveys [4] [31] [39] [40] .

Our emphasis here is on cluster algebras with coefficients. More precisely, we investigate certain symmetric cluster algebras of geometric type with coefficients. Coefficients are of great importance both in geometric examples of cluster algebras [27] [28] [8] [41] [23] and in the study of duality phenomena [18] as shown in [21]. Following [2], we consider two types of categories which allow us to incorporate coefficients into the representation-theoretic model:

1) 2-Calabi-Yau Frobenius categories;

2) 2-Calabi-Yau ‘subtriangulated’ categories, i.e. full subcategories of the form ^{⊥}(ΣD) of a 2-Calabi-Yau triangulated category C, where D is a rigid functorially finite subcategory of C and Σ the suspension functor of C.

In both cases, we establish the link between the category and its associated cluster algebra using (variants of) cluster characters in the sense of Palu [36]. For subtriangulated categories, we use the restriction of the cluster characters constructed in [36]. For Frobenius categories, we construct a suitable variant in section 3 (Theorem 3.3).

The work of Geiss-Leclerc-Schröer [26] [23] and Buan-Iyama-Reiten-Scott [2] provides us with large classes of 2-Calabi-Yau Frobenius categories and of 2-Calabi-Yau subtriangulated categories which admit cluster structures in the sense of [2]. Our general results imply that for these classes, the 2-Calabi-Yau categories do yield ‘categorifications’ of the corresponding cluster algebras with coefficients (Theorems 5.4 and 6.3). As an application, we show that Conjectures 7.2, 7.10 and 7.12 of [21] hold for these cluster algebras (Proposition 5.5 and Theorem 6.3). Let us recall the statements of these conjectures:

7.2 cluster monomials are linearly independent; 7.10 different cluster monomials have different g-vectors and the g-vectors of the cluster

Date: October 16, 2007. Last modified on December 28, 2008.

1991 Mathematics Subject Classification. 18E30, 16D90, 18G10.

Key words and phrases. Cluster algebra, Tilting, 2-Calabi-Yau category.

variables in any given cluster form a basis of the ambient lattice; 7.12 the g-vectors of a cluster variable with respect to two neighbouring clusters are

related by a certain piecewise linear transformation (so that the g-vectors equal the g^{†}-vectors of [13]).

In the case of cluster algebras with principal coefficients admitting a categorification by a 2-Calabi-Yau subtriangulated category, we obtain a representation-theoretic interpretation of the F -polynomial defined in section 3 of [21], cf. Theorem 6.5. This interpretation implies in particular that Conjecture 5.4 of [21] holds in this case: The constant coefficient of the F -polynomial equals 1. We also deduce that the multidegree of the F -polynomial associated with a rigid indecomposable equals the dimension vector of the corresponding module (Proposition 6.6). By combining this with recent work by Buan-Marsh-Reiten [5], cf. also [17], we obtain a counterexample to Conjecture 7.17 (and 6.11) of [21]. We point out that the corresponding computations were already present in G. Cerulli’s work [12]. Following a suggestion by A. Zelevinsky, we show that, by assuming the existence of suitable categorifications, instead of the equality claimed in Conjecture 7.17, one does have an inequality: The multidegree of the F -polynomial is greater or equal to the denominator vector (Proposition 6.8). We also show in certain cases that the transformation rule for g-vectors predicted by Conjecture 6.10 of [21] does hold (Proposition 6.9).

Let us emphasize that our proofs for certain cluster algebras of some of the conjectures of [21] depend on the existence of suitable Hom-finite 2-Calabi-Yau categories with a clustertilting object. This hypothesis imposes a finiteness condition on the corresponding cluster algebra (to the best of our knowledge, it is not known how to express this condition in combinatorial terms). The construction of such 2-Calabi-Yau categories is a non trivial problem for which we rely on [6] in the acyclic case and on [26] [23] [2] and [1] in the non acyclic case. As A. Zelevinsky has kindly informed us, many of the conjectures of [21] will be proved in [16] in full generality building on [35] and [15].

Acknowledgments. We thank A. Zelevinsky for stimulating discussions and for informing us about the ongoing work in [16] and [12]. We are grateful to A. Buan and I. Reiten for sharing the results on dimension vectors obtained in [5]. We thank J. Schröer and O. Iyama for their interest and for motivating discussions. The first-named author gratefully acknowledges a 5-month fellowship of the Liegrits network (MRTN-CT 2003-505078) during which this research was carried out. Both authors thank the referee for his great help in making this article more readable.

2. Recollections

2.1. Cluster algebras. In this section, we recall the construction of cluster algebras of geometric type with coefficients from [21]. For an integer x, we use the notations

[x]_{+} = max(x, 0)

and

sgn(x) = \{ \begin{matrix}−1 & if x < 0; \\ 0 & if x = 0; \\ −1 & if x < 0. \end{matrix}

The tropical semifield on a finite family of variables u_{j}, j ∈ J, is the abelian group (written multiplicatively) freely generated by the u_{j}, j ∈ J, endowed with the addition ⊕ defined by

∏_{j} u_{j}^{a_{j}} ⊕ ∏_{j} u_{j}^{b_{j}} = ∏_{j} u_{j}^{min(a_{j},b_{j})}.

Let 1 ≤ r ≤ n be integers. Let ℙ be the tropical semifield on the indeterminates x_{r+1}, …, x_{n}. Let ℚℙ be the group algebra on the abelian group ℙ. It identifies with the algebra of Laurent polynomials with rational coefficients in the variables x_{r+1}, …, x_{n}. Let F be the field of fractions of the ring of polynomials with coefficients in ℚℙ in r indeterminates. A seed in F is a pair (B̃, x) formed by

• an n×r-matrix B̃ with integer entries whose principal part B (the submatrix formed by the first r rows) is antisymmetric;

• a free generating set x = {x_{1}, …, x_{r}} of the field F. The matrix B̃ is called the exchange matrix and the set x the cluster of the seed (B̃, x). Let 1 ≤ s ≤ r be an integer. The seed mutation in the direction s transforms the seed (B̃, x) into the seed μ_{s}(B̃, x) = (B̃′, x′), where

• the entries b′_{ij} of B̃′ are given by

b′_{ij} = \{ \begin{matrix}−b_{ij} & if i = s or j = s; \\ b_{ij} + sgn(b_{is})[b_{is}b_{sj}]_{+} & otherwise. \end{matrix}

• The cluster x′ = {x′_{1}, … x′_{r}} is given by x′_{j} = x_{j} for j ≠ s whereas x′_{s} ∈ F is determined by the exchange relation

x′_{s}x_{s} = ∏_{i=1}^{n} x_{i}^{[b_{is}]_{+}} + ∏_{i=1}^{n} x_{i}^{[−b_{is}]_{+}}.

Mutation in a fixed direction is an involution.

Let 𝕋_{r} be the r-regular tree, whose edges are labeled by the numbers 1, …, r so that the r edges emanating from each vertex carry different labels. A cluster pattern is the assignment of a seed (B̃_{t}, x_{t}) to each vertex t of 𝕋_{r} such that the seeds assigned to vertices t and t′ linked by an edge labeled s are obtained from each other by the seed mutation μ_{s}.

Fix a vertex t_{0} of the r-regular tree 𝕋_{r}. Clearly, a cluster pattern is uniquely determined by the initial seed (B̃_{t_{0}}, x_{t_{0}}), which can be chosen arbitrarily.

Fix a seed (B̃, x) and let (B̃_{t}, x_{t}), t ∈ 𝕋_{r} be the unique cluster pattern with initial seed (B̃, x). The clusters associated with (B̃, x) are the sets x_{t}, t ∈ 𝕋_{r}. The cluster variables are the elements of the clusters. The cluster algebra A(B̃) = A(B̃, x) is the ℤℙ-subalgebra of F generated by the cluster variables. Its ring of coefficients is ℤℙ. It is a ‘cluster algebra without coefficients’ if r = n and thus ℤℙ = ℤ.

2.2. Cluster algebras from ice quivers. As we have seen in the previous subsection, our cluster algebras are given by certain integer matrices B̃. Such matrices can also be encoded by ‘ice quivers’: A quiver is a quadruple Q = (Q_{0}, Q_{1}, s, t), where Q_{0} is a set (the set of vertices), Q_{1} is a set (the set of arrows) and s and t are two maps Q_{1} → Q_{0} (taking an arrow to its source respectively to its target). An ice quiver is a pair (Q, F) consisting of a quiver Q and a subset F of its set of vertices (F is the set of frozen vertices).

Let (Q, F) be an ice quiver such that the set Q_{0} is the set of natural numbers from 1 to n, the set Q_{1} is finite and the set F is the set of natural numbers from r + 1 to n for some 1 ≤ r ≤ n. The associated integer matrix B̃(Q, F) is the n × r matrix whose entry b_{ij} equals the number of arrows from i to j minus the number of arrows from j to i. The cluster algebra with coefficients A(Q, F) is defined as the cluster algebra A(B̃(Q, F)). The matrix B̃(Q, F) determines the ice quiver (Q, F) if

1) Q does not have loops (arrows from a vertex to itself) and

2) Q does not have 2-cycles (pairs of distinct arrows α, β such that s(α) = t(β) and t(α) = s(β)) and

3) there are no arrows between any vertices of F.

Given integers 1 ≤ r ≤ n each integer matrix B̃ with antisymmetric principal part B (formed by the first r rows of B̃), is obtained as the matrix associated with a unique ice quiver satisfying these properties. The mutation of ice quivers satisfying conditions 1)-3) is defined via the mutation of the corresponding integer matrices recalled in section 2.1.

2.3. Krull-Schmidt categories. An additive category has split idempotents if each idempotent endomorphism e of an object X gives rise to a direct sum decomposition Y ⊕Z →^{∼} X such that Y is a kernel for e. A Krull-Schmidt category is an additive category where the endomorphism rings of indecomposable objects are local and each object decomposes into a finite direct sum of indecomposable objects (which are then unique up to isomorphism and permutation). Each Krull-Schmidt category has split idempotents. We write indec(C) for the set of isomorphism classes of indecomposable objects of a Krull-Schmidt category C.

Let C be a Krull-Schmidt category. An object X of C is basic if every indecomposable of C occurs with multiplicity ≤ 1 in X. In this case, X is fully determined by the full additive subcategory add(X) whose objects are the direct factors of finite direct sums of copies of X. The map X ↦ add(X) yields a bijection between the isomorphism classes of basic objects and the full additive subcategories of C which are stable under taking direct factors and only contain finitely many indecomposables up to ismorphism.

Let k be an algebraically closed field. A k-category is a category whose morphism sets are endowed with structures of k-vector spaces such that the composition maps are bilinear. A k-category is Hom-finite if all of its morphism spaces are finite-dimensional. A k-linear category is a k-category which is additive. Let C be a k-linear Hom-finite category with split idempotents. Then C is a Krull-Schmidt category. Let T be an additive subcategory of C stable under taking direct factors. The quiver Q = Q(T ) of T is defined as follows: The vertices of Q are the isomorphism classes of indecomposable objects of T and the number of arrows from the isoclass of T_{1} to that of T_{2} equals the dimension of the space of irreducible morphisms

irr(T_{1}, T_{2}) = rad(T_{1}, T_{2})/rad^{2}(T_{1}, T_{2}) ,

where rad denotes the radical of T , i.e. the ideal such that rad(T_{1}, T_{2}) is formed by all non isomorphisms from T_{1} to T_{2}.

2.4. 2-Calabi-Yau triangulated categories. Let k be an algebraically closed field. Let C be a k-linear triangulated category with suspension functor Σ. We assume that C is Hom-finite and has split idempotents. Thus, it is a Krull-Schmidt category. For objects X, Y of C and an integer i, we define

Ext^{i}(X, Y) = C(X, Σ^{i}Y).

An object X of C is rigid if Ext^{1}(X, X) = 0.

Let d be an integer. Following [42], cf. also [32], we define the category C to be d-Calabi-Yau if there exists a family of linear forms

t_{X} : C(X, Σ^{d}X) → k , X ∈ obj(C) ,

such that the bilinear forms

⟨, ⟩ : C(Y, Σ^{d}X) × C(X, Y) → k , (f, g) ↦ t_{X}(f ∘ g)

are non degenerate and satisfy

⟨Σ^{p}f, g⟩ = (−1)^{pq}⟨Σ^{q}g, f⟩

for all f in C(Y, Σ^{q}X) and all g ∈ C(X, Σ^{p}Y), where p + q = d.

Let us assume that C is 2-Calabi-Yau. A cluster-tilting subcategory of C is a full additive subcategory T ⊂ C which is stable under taking direct factors and such that

• for each object X of C, the functors C(X,?) : T → mod k and C(?, X) : T^{op} → mod k are finitely generated;

• an object X of C belongs to T iff we have Ext^{1}(T, X) = 0 for all objects T of T . A cluster-tilting object is a basic object T of C such that add(T) is a cluster-tilting

subcategory. Equivalently, an object T is cluster-tilting if it is rigid and if each object X satisfying Ext^{1}(T, X) = 0 belongs to add(T). The following definition is taken from section 1 of [2]. Recall that C is a Hom-finite k-linear triangulated category with split idempotents which is 2-Calabi-Yau.

Definition 2.1 ([2]). The cluster-tilting subcategories of C determine a cluster structure on C if the following hold:

0) There is at least one cluster-tilting subcategory in C.

1) For each cluster-tilting subcategory T′ of C and each indecomposable M of T′, there is a unique (up to isomorphism) indecomposable M^{∗} not isomorphic to M and such that the additive subcategory T′′ = μ_{M}(T′) of C with set of indecomposables

indec(T′′) = indec(T′) \ {M} ∪ {M^{∗}}

is a cluster-tilting subcategory;

2) In the situation of 1), there are triangles

\xymatrix{M^{∗} \ar[r]^{f} & E \ar[r]^{g} & M \ar[r] & ΣM^{∗}} and \xymatrix{M \ar[r]^{s} & E′ \ar[r]^{t} & M^{∗} \ar[r] & ΣM^{∗}} ,

where g and t are minimal right T′ ∩ T′′-approximations and f and s are minimal left T′ ∩ T′′-approximations.

3) For any cluster-tilting subcategory T′, the quiver Q(T′) does not have loops nor 2-cycles.

4) We have Q(μ_{M}(T′)) = μ_{M}(Q(T′)) for each cluster-tilting subcategory T′ of C and

each indecomposable M of T′. The cluster tilting subcategory T′′ = μ_{M}(T′) of 1) is the mutation of T′ at the indecomposable object M. The mutation of a cluster-tilting object T is defined via the mutation of the cluster-tilting subcategory add(T).

Lemma 2.2. Suppose that the cluster-tilting subcategories determine a cluster structure on C. Then, in the situation of condition 2) of the above definition 2.1, the following hold:

a) The space Ext^{1}(M, M^{∗}) is one-dimensional (hence, by the 2-Calabi-Yau property, so is the space Ext^{1}(M^{∗}, M)) and the triangles of 2) are non split.

3) The multiplicity of an indecomposable U of T′ ∩ T′′ in E equals the number of arrows from U to M in the quiver Q(T′) and that from M^{∗} to U in Q(T′′); the multiplicity of U in E′ equals the number of arrows from M to U in Q(T′) and that from U to M^{∗} in Q(T′′);

Proof. a) The first triangle yields an exact sequence

C(M, E) → C(M, M) → Ext^{1}(M, M^{∗}) → 0.

By the absence of loops required in condition 3), each radical morphism from M to M factors through E. Since k is algebraically closed, the radical is of codimension 1 in the local algebra C(M, M). Thus, the space Ext^{1}(M, M^{∗}) is one-dimensional. The minimality of the approximations implies that the triangles are non split. b) This follows if we combine the definition of the quivers Q(T′) and Q(T′′), with the approximation properties of f, g, s and t. □

We refer to section 1, page 11 of [2] for a list of classes of examples where this assumption holds. In particular, this list contains the cluster categories associated with finite quivers without oriented cycles and the stable categories of preprojective algebras of Dynkin quivers. We refer the reader to the surveys [4] [39] [30] [31] for more information on cluster categories and to the survey [24] for more information on stable categories of Dynkin quivers.

2.5. Cluster characters. The notion of cluster character is due to Palu [37]. Under suitable assumptions, cluster characters allow one to pass from 2-Calabi-Yau categories to cluster algebras. We recall the definition and the main construction from [37]. Let k be an algebraically closed field and C a k-linear Hom-finite triangulated category with split idempotents which is 2-Calabi-Yau. Let R be a commutative ring. A cluster character on C with values in R is a map ζ : obj(C) → R such that

1) we have ζ(L) = ζ(L′) if and L and L′ are isomorphic,

2) we have ζ(L ⊕ M) = ζ(L)ζ(M) for all objects L and M and

3) if L and M are objects such that Ext^{1}(L, M) is one-dimensional (and hence Ext^{1}(M, L) is one-dimensional) and

L → E → M → ΣL and M → E′ → L → ΣM

are non-split triangles, then we have

(2.1)  ζ(L)ζ(M) = ζ(E) + ζ(E′).

Assume that C has a cluster-tilting object T which is the direct sum of r pairwise non isomorphic indecomposable summands T_{1}, … T_{r}. In a vast generalization of Caldero-Chapoton’s work [9], Palu has shown in [37] that there is a canonical cluster-character

X_{?}^{T} : obj(C) → ℤ[x_{1}, …, x_{r}] , M ↦ X_{M}^{T}

such that X_{ΣT_{i}}^{T} = x_{i} for 1 ≤ i ≤ r. Let us recall Palu’s construction. First we need to introduce some more notation. Let C be the endomorphism algebra of T. Let mod C denote the category of k-finite-dimensional right C-modules. For each 1 ≤ i ≤ r, the morphism space C(T, T_{i}) becomes an indecomposable projective right C-module denoted by P_{i}. Its simple top is denoted by S_{i}. For L and M in mod C, define

⟨L, M⟩ = dim Hom_{C}(L, M) − dim Ext_{C}^{1}(L, M)

and put

⟨L, M⟩_{a} = ⟨L, M⟩ − ⟨M, L⟩.

By Theorem 11 of [37], the map (L, M) ↦ ⟨L, M⟩_{a} induces a well-defined bilinear form on the Grothendieck group K_{0}(mod C). By [34], for any X ∈ C, we have triangles

T_{1}^{X} → T_{0}^{X} → X → ΣT_{1}^{X} and X → Σ^{2}T_{X}^{0} → Σ^{2}T_{X}^{1} → ΣX,

where T_{1}^{X}, T_{0}^{X}, T_{X}^{0} and T_{X}^{1} belong to add(T). The index and coindex of X with respect to T are defined to be the classes in K_{0}(add T)

ind_{T} (X) = [T_{0}^{X}] − [T_{1}^{X}] and coind_{T} (X) = [T_{X}^{0}] − [T_{X}^{1}].

For an object M of C, one defines

X_{M}^{T} = ∏_{i=1}^{r} x_{i}^{−[coind_{T} (M):T_{i}]} ∑_{e} χ(Gr_{e}(C(T, M)) ∏_{i=1}^{r} x_{i}^{⟨S_{i},e⟩_{a}},

where e runs through the positive elements of the Grothendieck group K_{0}(mod C) and Gr_{e}(C(T, M)) denotes the variety of submodules U of the right C-module C(T, M) such that the class of U is e and χ is the Euler characteristic (of the underlying topological space if k = ℂ or of l-adic cohomology if k is arbitrary).

2.6. From 2-CY categories to cluster algebras without coefficients. In this section, we show how certain 2-Calabi-Yau triangulated categories can be linked to cluster algebras without coefficients via cluster characters. All we need to do is to combine the results recalled in sections 2.4 and 2.5. In the main part of the paper, we will concentrate on the case where our cluster algebras do have coefficients.

Let k be an algebraically closed field and C a Hom-finite k-linear 2-Calabi-Yau triangulated category with split idempotents as defined in section 2.4. Let T be a cluster-tilting object in C. Assume that T is the direct sum of r pairwise non isomorphic indecomposable objects T_{1}, …, T_{r}. Let

ζ_{T} : obj(C) → ℚ(x_{1}, …, x_{r})

be Palu’s cluster character associated with T as recalled in section 2.5. In particular, we have

(2.2)  ζ_{T} (ΣT_{i}) = x_{i} for 1 ≤ i ≤ r.

Now assume that the cluster-tilting subcategories define a cluster structure on C (cf. section 2.4). A cluster-tilting object T′ is reachable from T if add(T′) is obtained from add(T) be a finite sequence of mutations as defined in 2.4. A rigid object M is reachable from T if it lies in add(T′) for a cluster-tilting object T′ reachable from T. Let Q be the quiver of the endomorphism algebra C of T, or equivalently, the quiver of the category add(T). We consider Q as an ice quiver with empty set of frozen vertices and denote by A(Q) the associated cluster algebra without coefficients (defined by specializing the construction of 2.2 to the case where the set of frozen vertices is empty). It is the subalgebra of ℚ(x_{1}, …, x_{r}) generated by the cluster variables.

Proposition 2.3. Assume that the above assumptions hold and in particular that the cluster-tilting subcategories define a cluster-structure on C (cf. section 2.4).

a) The map M ↦ ζ_{T} (ΣM) induces a surjection from the set of rigid objects reachable from T onto the set of cluster variables of the cluster algebra A(Q).

b) The surjection of a) induces a surjection from the set of cluster-tilting objects reachable from T onto the set of clusters of A(Q).

Proof. Clearly, part a) follows from part b). Let us prove part b). Let 𝕋_{r} be the r-regular tree and let t_{0} be a fixed vertex of 𝕋_{r}. Let B be the antisymmetric matrix associated with the quiver Q and let x be the initial cluster x_{1}, …, x_{r}. Let (B_{t}, x_{t}), t ∈ 𝕋_{r}, be the unique cluster pattern with initial seed (B_{t_{0}}, x_{t_{0}}) = (B, x) (cf. section 2.1). To each vertex t of 𝕋, we assign a cluster-tilting object T_{t} with indecomposable direct summands T_{t,1}, …, T_{t,r} such that

1) we have T_{t_{0}} = T and

2) if t is linked to t′ by an edge labeled s, then T_{t′} is obtained from T_{t} by mutating at

the summand T_{t,s}. It follows from point 1) of the definition of a cluster structure that T_{t} is well-defined for each vertex t of 𝕋. Moreover, it follows from point 4) of the same definition that for each vertex t of 𝕋, the antisymmetric matrix B_{t} corresponds to the quiver of the category add(T_{t}) under the bijection of section 2.2. We claim that for each vertex t of 𝕋, the cluster character takes the shift ΣT_{t,i} of the indecomposable direct summand T_{t,i} of T_{t} to the cluster variable x_{t,i}, 1 ≤ i ≤ r. Indeed, this holds for t = t_{0} by equation (2.2). Now assume that it holds for some vertex t and that t is linked to a vertex t′ by an edge labeled s. We know that the indecomposable summands of T_{t′} are the T_{t′,i} = T_{t,i} for i ≠ s and a new summand T′_{t,s} which is not isomorphic to T_{t,s}. By part a) of lemma 2.2, the extension space between T_{t,s} and T_{t′,s} is one-dimensional and we have non split triangles

T_{t′,s} → E → T_{t,s} → ΣT_{t′,s} and T_{t,s} → E′ → T_{t′,s} → ΣT_{t,s}.

Here, the middle terms are sums of copies of the T_{t,i}, i ≠ s, and the multiplicities are determined by the quivers of the endomorphism algebras of T and T′, as indicated in part b) of lemma 2.2. More precisely, if b_{ij}^{t} denotes the (i, j)-entry of the exchange matrix, the summand T_{t,i} occurs in E with multiplicity [b_{is}^{t}]_{+} and in E′ with multiplicity [b_{si}^{t}]_{+} = [−b_{is}^{t}]_{+}. Now if we use points 2) and 3) of the definition of a cluster character, we see that the induction hypothesis and equation (2.1) yield the exchange relation

x_{t,s}ζ_{T} (ΣT_{t′,s}) = ∏_{i=1}^{n} x_{i}^{[b_{ik}^{t}]_{+}} + ∏_{i=1}^{n} x_{i}^{[−b_{ik}^{t}]_{+}}.

Thus, we have ζ_{T} (ΣT_{t′,s}) = x_{t′,s} as required. □

2.7. Frobenius categories. A Frobenius category is an exact category in the sense of Quillen [38] which has enough projectives and enough injectives and where an object is projective iff it is injective. By definition, such a category is endowed with a class of admissible exact sequences

0 → L → M → N → 0.

Following [22] we will call the left morphism L → M of such a sequence an inflation, the right morphism a deflation and, sometimes, the whole sequence a conflation. Let E be a Frobenius category. Its associated stable category \underline{E} is the quotient of E by the ideal of morphisms factoring through a projective-injective object. It was shown by Happel [29] that \underline{E} has a canonical structure of triangulated category. We have

Ext_{E}^{i}(L, M) →^{∼} Exti ; \underline{E}(L, M)

for all objects L and M of E and all integers i ≥ 1. An object M of E is rigid if Ext_{E}^{1}(M, M) = 0.

Let k be an algebraically closed field and E a Hom-finite Frobenius category with split idempotents. Suppose that E is a 2-Calabi-Yau Frobenius category, i.e. its associated stable category C = \underline{E} is 2-Calabi-Yau in the sense of section 2.4. A cluster-tilting subcategory of E is a full additive subcategory T ⊂ E which is stable under taking direct factors and such that

• for each object X of E, the functors E(X,?) : T → mod k and E(?, X) : T^{op} → mod k are finitely generated;

• an object X of E belongs to T iff we have Ext_{E}^{1}(T, X) = 0 for all objects T of T . Clearly if these conditions hold, each projective-injective object of E belongs to T . A cluster-tilting object is a basic object T of E such that add(T) is a cluster-tilting subcategory. Equivalently, an object T is cluster-tilting if it is rigid and if each object X satisfying Ext_{E}^{1}(T, X) = 0 belongs to add(T). The following definition is taken from section 1 of [2]. Recall that E is a k-linear Hom-finite Frobenius category with split idempotents such that the associated stable category C = \underline{E} is 2-Calabi-Yau.

Definition 2.4 ([2]). The cluster-tilting subcategories of E determine a cluster structure on E if the following hold:

0) There is at least one cluster-tilting subcategory in E.

1) For each cluster-tilting subcategory T′ of E and each non projective indecomposable M of T′, there is a unique (up to isomorphism) non projective indecomposable M^{∗} not isomorphic to M and such that the additive subcategory T′′ = μ_{M}(T′) of E with set of indecomposables

indec(T′′) = indec(T′) \ {M} ∪ {M^{∗}}

is a cluster-tilting subcategory;

2) In the situation of 1), there are conflations

\xymatrix{0 \ar[r] & M^{∗} \ar[r]^{f} & E \ar[r]^{g} & M \ar[r] & 0} and \xymatrix{0 \ar[r] & M \ar[r]^{s} & E′ \ar[r]^{t} & M^{∗} \ar[r] & 0} ,

where g and t are minimal right T′ ∩ T′′-approximations and f and s are minimal left T′ ∩ T′′-approximations.

3) For any cluster-tilting subcategory T′, the quiver Q(T′) does not have loops nor 2-cycles.

4) We have Q^{∘}(μ_{M}(T′)) = μ_{M}(Q^{∘}(T′)) for each cluster-tilting subcategory T′ of E and each non projective indecomposable M of T′, where Q^{∘}(T′) denotes the quiver obtained from Q(T′) be removing all arrows between projective vertices.

The cluster tilting subcategory T′′ = μ_{M}(T′) of 1) is the mutation of T′ at the non projective indecomposable object M. The mutation of a cluster-tilting object T is defined via the mutation of the cluster-tilting subcategory add(T).

Lemma 2.5. Suppose that the cluster-tilting subcategories determine a cluster structure on E. Then, in the situation of condition 2) of the above definition 2.4, the following hold:

a) The space Ext^{1}(M, M^{∗}) is one-dimensional (hence, by the 2-Calabi-Yau property, so is the space Ext^{1}(M^{∗}, M)) and the conflations of 2) are non split.

b) The multiplicity of an indecomposable U of T′ ∩ T′′ in E equals the number of arrows from U to M in the quiver Q(T′) and that from M^{∗} to U in Q(T′′); the multiplicity of U in E′ equals the number of arrows from M to U in Q(T′) and that from U to M^{∗} in Q(T′′);

We omit the proof of the lemma since it is entirely parallel to that of lemma 2.2. Large classes of examples of Frobenius categories where the cluster-tilting objects define a clusterstructure are obtained in [23] and [3], cf. the survey [24] and example 5.3 below. For an extension of the theory from the antisymmetric to the antisymmetrizable case, we refer to [14].

3. Cluster characters for 2-Calabi-Yau Frobenius categories

Let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents. We assume that E is Hom-finite and that the stable category C = \underline{E} is 2-Calabi-Yau (cf. section 2.4).

Definition 3.1. A cluster character on E with values in a commutative ring R is a map ζ : obj(E) → R such that

1) we have ζ(L) = ζ(L′) if and L and L′ are isomorphic,

2) we have ζ(L ⊕ M) = ζ(L)ζ(M) for all objects L and M and

3) if L and M are objects such that Ext_{E}^{1}(L, M) is one-dimensional (and hence Ext_{E}^{1}(M, L) is one-dimensional) and

0 → L → E → M → 0 and 0 → M → E′ → L → 0

are non-split triangles, then we have

(3.1)  ζ(L)ζ(M) = ζ(E) + ζ(E′).

From now on, we assume in addition that E contains a cluster-tilting object T. Using T we will construct a cluster character on E and link it to Palu’s cluster character associated with the image of T in the triangulated category C = \underline{E} (cf. section 2.5).

Let C be the endomorphism algebra of T (in E) and \underline{C} = End_{C}(T). Let

F = Hom_{E}(T,?) : E → mod C,
G = Hom_{C}(T,?) : C → mod \underline{C}.

Let T_{i}, 1 ≤ i ≤ n, be the pairwise non isomorphic indecomposable direct summands of T. We choose the numbering of the T_{i} so that T_{i} is projective exactly for r < i ≤ n for some integer 1 ≤ r ≤ n. For 1 ≤ i ≤ n, let S_{i} be the top of the indecomposable projective P_{i} = F T_{i}. Note that C and \underline{C} are finite dimensional k-algebras, so finitely presented modules are the same as finitely generated modules. As in section 4 of [34], we identify Mod \underline{C} with the full subcategory of Mod C formed by the modules without composition factors isomorphic to one of the S_{i}, r < i ≤ n. Let DC be the unbounded derived category of the abelian category Mod C of all right C-modules. Let per C be the perfect derived category of C, i.e. the full subcategory of DC whose objects are all the complexes quasi-isomorphic to bounded complexes of finitely generated projective C-modules. Let D^{b}(mod C) the bounded derived category of mod C identified with the full subcategory of DC whose objects are all complexes whose total homology is finite-dimensional over k. As shown in section 4 of [34], we have the following embeddings

mod \underline{C} ↪ per C ↪ D^{b}(mod C).

We have a bilinear form

⟨ , ⟩ : K_{0}(per C) × K_{0}(D^{b}(mod C)) ⟶ ℤ

defined by

⟨[P], [X]⟩ = ∑(−1)^{i}dim Hom_{D^{b}(mod C)}(P, Σ^{i}X),

where K_{0}(per C) (resp. K_{0}(D^{b}(mod C))) is the Grothendieck group of per C (resp. D^{b}(mod C)) and Σ is the shift functor of D^{b}(mod C).

For arbitrary finitely generated C-modules L and N, put

[L, N] = ^{0}[L, N] = dim_{k} Hom_{C}(L, N) and ^{i}[L, N] = dim_{k} Ext_{C}^{i}(L, N) for i ≥ 1.

Let

⟨L, N⟩_{τ} = [L, N] − ^{1}[L, N] and ⟨L, N⟩_{3} = ∑_{i=0}^{3} (−1)^{i} ^{i}[L, N]

be the truncated Euler forms on the split Grothendieck group K_{0}^{sp}(mod C). By the proposition below, if L is a \underline{C}-module, then ⟨L, N⟩_{3} only depends on the dimension vector \underline{dim} L in K_{0}(mod C). We put

⟨\underline{dim} L, N⟩_{3} = ⟨L, N⟩_{3}.

Proposition 3.2. a) The restriction of the map

K_{0}(per C) ⟶ K_{0}(D^{b}(mod C)) = K_{0}(mod C)

induced by the inclusion of per C into D^{b}(mod C) to the subgroup generated by the [S_{i}], 1 ≤ i ≤ r, is injective.

b) If L, N are two \underline{C}-modules such that \underline{dim} L = \underline{dim} N in K_{0}(mod C), then

⟨L, Y ⟩_{3} = ⟨N, Y ⟩_{3}

for each finitely generated C-module Y.

Proof. a) We need to show that for arbitrary finitely generated \underline{C}-modules L, N with \underline{dim} L = \underline{dim} N, we have [L] = [N] in K_{0}(per C). Let

0 = L_{s} ⊂ L_{s−1} ⊂ ⋯ ⊂ L_{0} = L

and

0 = N_{s} ⊂ N_{s−1} ⊂ ⋯ ⊂ N_{0} = N

be composition series of L and N respectively. By [34], we know that every \underline{C}-module has projective dimension at most 3 in mod C. Assume for simplicity that L_{s−1} = S_{1}, L_{s−2}/L_{s−1} = S_{2}. Denote by P_{i}^{∗} a minimal projective resolution of S_{i}. Then we have the following commutative diagram

\xymatrix{0 \ar[r] & P_{1}^{3} \ar[r] \ar[d] & P_{1}^{2} \ar[r] \ar[d] & P_{1}^{1} \ar[r] \ar[d] & P_{1}^{0} \ar[r] \ar[d] & L_{s−1} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & P_{1}^{3} ⊕ P_{2}^{3} \ar[r] \ar[d] & P_{1}^{2} ⊕ P_{2}^{2} \ar[r] \ar[d] & P_{1}^{1} ⊕ P_{2}^{1} \ar[r] \ar[d] & P_{1}^{0} ⊕ P_{2}^{0} \ar[r] \ar[d] & L_{s−2} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & P_{2}^{3} \ar[r] & P_{2}^{2} \ar[r] & P_{2}^{1} \ar[r] & P_{2}^{0} \ar[r] & L_{s−2}/L_{s−1} \ar[r] & 0}

where the middle term is a projective resolution of L_{s−2}. In this way, we inductively construct projective resolutions for L and N. If m_{i} is the multiplicity of S_{i} in the composition factors of L and N, then we obtain projective resolutions of L and N of the form

0 → ⨁_{i=1}^{r}(P_{i}^{3})^{m_{i}} ⟶^{f_{3}} ⨁_{i=1}^{r}(P_{i}^{2})^{m_{i}} ⟶^{f_{2}} ⨁_{i=1}^{r}(P_{i}^{1})^{m_{i}} ⟶^{f_{1}} ⨁_{i=1}^{r}(P_{i}^{0})^{m_{i}} → L → 0,

0 → ⨁_{i=1}^{r}(P_{i}^{3})^{m_{i}} ⟶^{g_{3}} ⨁_{i=1}^{r}(P_{i}^{2})^{m_{i}} ⟶^{g_{2}} ⨁_{i=1}^{r}(P_{i}^{1})^{m_{i}} ⟶^{g_{1}} ⨁_{i=1}^{r}(P_{i}^{0})^{m_{i}} → N → 0.

Let P^{L} (resp. P^{N}) be the projective resolution complex of L (resp. N). We have L ≅ P^{L} and N ≅ P^{N} in per C, which implies [L] = [P^{L}] = [P^{N}] = [N] in K_{0}(per C).

b) We have

⟨L, Y ⟩_{3} = ⟨P^{L}, Y ⟩ = ⟨[P^{L}], [Y]⟩,
⟨N, Y ⟩_{3} = ⟨P^{N}, Y ⟩ = ⟨[P^{N}], [Y]⟩.

By a), we have [P^{L}] = [P^{N}] in K_{0}(per C), which implies the equality. □

One should note that the truncated Euler form ⟨ , ⟩_{3} does not descend to the Grothendieck group K_{0}(mod C) in general (except if the global dimension of C is not greater than 3), cf. remark 3.5.

Using the bilinear forms introduced so far, for M ∈ E, we define the Laurent polynomial

X′_{M} = ∏_{i=1}^{n} x_{i}^{⟨F M,S_{i}⟩_{τ}} ∑_{e} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{i=1}^{n} x_{i}^{−⟨e,S_{i}⟩_{3}}.

Here we consider Ext_{E}^{1}(T, M) as a right C-module via the natural action of C = End_{E}(T) on the first argument; the sum ranges over all the elements of the Grothendieck group; for a C-module L, the notation Gr_{e}(L) denotes the projective variety of submodules of L whose class in the Grothendieck group is e; for an algebraic variety V, the notation χ(V) denotes the Euler characteristic (of the underlying topological space of V if k = ℂ and of l-adic cohomology if k is arbitrary).

Since C is 2-Calabi-Yau, the object \underline{T} = ⊕_{i=1}^{r}T_{i} is a cluster tilting object of C. For an object M of C, put

X_{M} = X\underline{T} ; M ,

where M ↦ X\underline{T} ; M is Palu’s cluster character associated with the cluster-tilting object \underline{T}, cf. section 2.5.

The following theorem shows that M ↦ X′_{M} is a cluster character on E and that, if we specialize the ‘coefficients’ x_{r+1}, …, x_{n} to 1, it specializes to the composition of Palu’s cluster character M ↦ X_{M} with the suspension functor M ↦ ΣM. Notice that this theorem does not involve cluster algebras (but paves the way for establishing a link with cluster algebras when E admits a cluster structure, cf. Theorem 5.4 below).

Theorem 3.3. As above, let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents such that E is Hom-finite, the stable category C = \underline{E} is 2-Calabi-Yau and E contains a cluster-tilting object T. For an object M of E, let X′_{M} and X_{M} be the Laurent polynomials defined above.

a) We have X′_{T_{i}} = x_{i} for 1 ≤ i ≤ n.

b) The specialization of X′_{M} at x_{r+1} = x_{r+2} = … = 1 is X_{ΣM}, where Σ is the suspension of C.

c) For any two objects L and M of E, we have X′_{L⊕M} = X′_{L}X′_{M}.

d) If L and M are objects of E such that Ext_{E}^{1}(L, M) is one-dimensional and we have non split conflations

0 → L → E → M → 0 and 0 → M → E′ → L → 0 ,

then we have

X′_{L}X′_{M} = X′_{E} + X′_{E′}.

Proof. a) is straightforward.

b) We have

X_{ΣM} = ∏_{i=1}^{r} x_{i}^{−[coind\underline{T} (ΣM):T_{i}]} ∑_{e} χ(Gr_{e}(GΣM)) ∏_{i=1}^{r} x_{i}^{⟨S_{i},e⟩_{a}}.

Now by the definition, we have

GΣM = Hom_{C}(T, ΣM) = Ext_{E}(T, M).

Therefore, we only need to show that the exponents of x_{i}, 1 ≤ i ≤ r, in the corresponding terms of X_{ΣM} and X′_{M} are equal. There exists a triangle in C

T_{M}^{1} → T_{M}^{0} → M → ΣT_{1}

with T_{M}^{0} and T_{M}^{1} in add \underline{T}. We may and will assume that this triangle is minimal, i.e. does not admit a non zero direct factor of the form

T′ → T′ → 0 → ΣT′.

Since E is Frobenius, we can lift this triangle to a short exact sequence in E

0 → T_{M}^{1} → T_{M}^{0} ⊕ P → M → 0,

where P is a projective of E. Applying the functor F to this short exact sequence, we get a projective resolution of F M as a C-module,

0 → F T_{M}^{1} → F (T_{M}^{0} ⊕ P) → F M → 0.

Therefore, we have

⟨F M, S_{i}⟩_{τ} = [F T_{M}^{0} ⊕ F P, S_{i}] − [F T_{M}^{1}, S_{i}] = [F T_{M}^{0}, S_{i}] − [F T_{M}^{1}, S_{i}]

for 1 ≤ i ≤ r.

On the other side, we have the following minimal triangle

ΣM → Σ^{2}T_{M}^{1} → Σ^{2}T_{M}^{0} → Σ^{2}M.

By the definition of the coindex, we get

−[coind\underline{T} (ΣM) : T_{i}] = −[T_{M}^{1} − T_{M}^{0} : T_{i}] = ⟨F M, S_{i}⟩_{τ} , for 1 ≤ i ≤ r.

Next we will show that ⟨S_{i}, e⟩_{a} = −⟨e, S_{i}⟩_{3}. Let N be a C-module such that \underline{dim} N = e. Note that N and the S_{i}, 1 ≤ i ≤ r, are \underline{C}-modules and that all of them are finitely presented C-modules. Therefore, they lie in the perfect derived category per(C). Thus, we can use the relative 3-Calabi-Yau property of per(C) (cf. [34]) to deduce that ⟨S_{i}, e⟩_{a} = −⟨e, S_{i}⟩_{3}. We have

Ext_{C}^{2}(N, S_{i}) = Ext_{C}(S_{i}, N) = Ext\underline{C}(S_{i}, N),
Ext_{C}^{3}(N, S_{i}) = Hom_{C}(S_{i}, N) = Hom\underline{C}(S_{i}, N),

for 1 ≤ i ≤ r. By the definition of ⟨S_{i}, N⟩_{a}, we have

⟨S_{i}, N⟩_{a} = dim_{k} Hom\underline{C}(S_{i}, N) − dim_{k} Ext\underline{C}(S_{i}, N) + dim_{k} Ext\underline{C}(N, S_{i}) − dim_{k} Hom\underline{C}(N, S_{i})
= dim_{k} Hom_{C}(S_{i}, N) − dim_{k} Ext_{C}(S_{i}, N) + dim_{k} Ext_{C}(N, S_{i}) − dim_{k} Hom_{C}(N, S_{i})
= ^{3}[N, S_{i}] − ^{2}[N, S_{i}] + ^{1}[N, S_{i}] − [N, S_{i}]
= −⟨N, S_{i}⟩_{3}.

c) is proved in exactly the same way as Corollary 3.7 in [9].

d) Let

0 → L ⟶^{i} E ⟶^{p} M → 0 and 0 → M ⟶^{i′} E′ ⟶^{p′} L → 0 ,

be the non-split conflations in E, and

ΣL ⟶^{GΣi} ΣE ⟶^{GΣp} ΣM → Σ^{2}L

ΣM ⟶^{GΣi′} ΣE′ ⟶^{GΣp′} ΣL → Σ^{2}N

the associated triangles in C. For any classes e, f, g in the Grothendieck group K_{0}(mod \underline{C}), let X_{e,f} be the variety whose points are the \underline{C}-submodules E ⊂ GΣE such that the dimension vector of (GΣi)^{−1}E equals e and the dimension vector of (GΣp)E equals f. Similarly, let Y_{f,e} be the variety whose points are the \underline{C}-submodules E ⊂ GΣE′ such that the dimension vector of (GΣi′)^{−1}E equals f and the dimension vector of (GΣp′)E equals e. Put

X_{e,f}^{g} = X_{e,f} ∩ Gr_{g}(GΣE),
Y_{f,e}^{g} = Y_{f,e} ∩ Gr_{g}(GΣE′).

Since C is a 2-CY triangulated category, by section 5.1 of [37] we also have

χ(Gr_{e}(GΣL) × Gr_{f}(GΣM)) = ∑_{g} χ(X_{e,f}^{g}) + χ(Y_{f,e}^{g}).

Therefore, part d) is a consequence of the following lemma.

Lemma 3.4. If X_{e,f}^{g} ≠ ∅, then we have the following equality

−⟨g, S_{i}⟩_{3} + ⟨F E, S_{i}⟩_{τ} = −⟨e + f, S_{i}⟩_{3} + ⟨F L, S_{i}⟩_{τ} + ⟨F M, S_{i}⟩_{τ} , 1 ≤ i ≤ n.

Proof. We have the following commutative diagram as in section 4 of [37]

\xymatrix{(GΣi)^{−1}E \ar[r]^{α} \ar[d]_{i} & E \ar[r]^{β} \ar[d]_{j} & (GΣp)E \ar[r] \ar[d]_{k} & 0 \\ GΣL \ar[r]^{GΣi} & GΣE \ar[r]^{GΣp} & GΣM \ar[r] & GΣ^{2}L}

where i, j, k are monomorphisms, β is an epimorphism and [E] = g, [GΣi)^{−1}E] = e, [GΣp)E] = f in K_{0}(mod C). One can easily show that ker GΣi = ker α. We have an exact sequence

0 → ker α → (GΣi)^{−1}E → E → (GΣp)E → 0.

If we apply F = Hom_{E}(T,?) to the short exact sequence

0 → L → E → M → 0,

we get the long exact sequences of C-modules

0 → F L → F E → F M → GΣL ⟶^{GΣi} GΣE → …,

and

0 → F L ⟶^{F i} F E ⟶^{F p} F M → ker α → 0.

Since ker α, (GΣi)^{−1}E, E, (GΣp)E are \underline{C}-modules, and the projective dimensions of F L, F E, F M are not greater than 1, we can use the method of Proposition 3.2 to construct the projective resolutions and compute the truncated Euler forms. We get that

⟨e, S_{i}⟩_{3} + ⟨f, S_{i}⟩_{3} = ⟨g, S_{i}⟩_{3} + ⟨ker α, S_{i}⟩_{3},

and

⟨F L, S_{i}⟩_{3} + ⟨F M, S_{i}⟩_{3} = ⟨F E, S_{i}⟩_{3} + ⟨ker α, S_{i}⟩_{3}.

Note that ⟨F L, S_{i}⟩_{3} = ⟨F L, S_{i}⟩_{τ}, ⟨F M, S_{i}⟩_{3} = ⟨F M, S_{i}⟩_{τ} and ⟨F E, S_{i}⟩_{3} = ⟨F E, S_{i}⟩_{τ}, which implies

⟨F L, S_{i}⟩_{τ} + ⟨F M, S_{i}⟩_{τ} − ⟨e + f, S_{i}⟩_{3} = ⟨F E, S_{i}⟩_{τ} − ⟨g, S_{i}⟩_{3}.
□

Remark 3.5. If C has finite global dimension, the Grothendieck group K_{0}(mod C) has the Euler form ⟨ , ⟩. We can then define a Laurent polynomial X_{M}^{f} as follows

X_{M}^{f} = ∏_{i=1}^{n} x_{i}^{⟨F M,S_{i}⟩} ∑_{e} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{i=1}^{n} x_{i}^{⟨S_{i},e⟩}.

One can show that in this case X′_{M} = X_{M}^{f}. In fact, if gldimC < ∞, then the perfect derived category per(C) equals D^{b}(mod C), and S_{i} belongs to per(C) for all i. Thus, we have

⟨S_{i}, e⟩ = ∑_{i=0}^{3}(−1)^{i}dim Ext_{C}^{i}(S_{i}, e) = −⟨e, S_{i}⟩_{3}

and ⟨F M, S_{i}⟩_{τ} = ⟨F M, S_{i}⟩. The assumption that C is of finite global dimension holds for the examples constructed in [2] by Proposition I.2.5 b) of [loc. cit.] and for the examples constructed in [26] by Proposition 11.5 of [loc.cit.].

4. Index and g-vector

4.1. Index. As in section 3, we let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents. We assume that E is Hom-finite and that the stable category C = \underline{E} is 2-Calabi-Yau (cf. section 2.4). Moreover, we assume that E admits a cluster-tilting object T and we write C = End_{E}(T) and \underline{C} = End_{C}(T).

Let D(Mod C) be the derived category of C-modules, D^{−}(mod C) the right bounded derived category of mod C, H^{−}(P) the right bounded homotopy category of finitely generated projective C-modules. It is well known that there is an equivalence

H^{−}(P) ⟶^{∼} D^{−}(mod C).

Proposition 4.1. For an arbitrary \underline{C}-module Z which is also a finitely presented C-module we have a canonical isomorphism

D Hom_{D^{−}(mod C)}(Z,?) ⟶^{∼} Hom_{D^{−}(mod C)}(?, Z[3]).

Proof. For arbitrary X ∈ D^{−}(mod C), by the equivalence, we have a P_{X} ∈ H^{−}(P) such that X ≅ P_{X} in D^{−}(mod C). Assume that P_{X} has the following form

… → P_{m} → P_{m+1} → … → P_{n−1} → P_{n} → 0 → 0 ….

Put

X_{0} = … → 0 → 0 → P_{n} → 0 …,
X_{i} = … → 0 → P_{n−i} → … → P_{n} → 0 …, for i > 0.

Clearly, the complex P_{X} is the direct limit of the complexes X_{i}. We write hocolim for the total left derived functor of the functor of taking the direct limit. Since taking direct limits over filtered systems is an exact functor, the functor hocolim is simply induced by the direct limit functor. Thus, we have P_{X} ≅ hocolim X_{i} in D(Mod C). Note that by Proposition 4 of [34], Z belongs to per C, i.e. Z is compact in D(Mod C). So we have

Hom_{D(Mod C)}(Z, X) ≅ Hom_{D(Mod C)}(Z, P_{X})
≅ Hom_{D(Mod C)}(Z, hocolimX_{i})
≅ colim Hom_{D(Mod C)}(Z, X_{i}).

By the definition of X_{i}, we know that X_{i} ∈ per C. Since per C is a full subcategory of D(Mod C), by the relative 3-Calabi-Yau property of per C, we have the following

colim Hom_{D(Mod C)}(Z, X_{i}) ≅ colimD Hom_{D(Mod C)}(X_{i}, Z[3]).

It is easy to see that this colimit is a stationary system, i.e. ∃ N such that for i > N, we have

D Hom_{D(Mod C)}(X_{i}, Z[3]) ≅ D Hom_{D(Mod C)}(X_{i+1}, Z[3]).

Thus, we have

colimD Hom_{D(Mod C)}(X_{i}, Z[3]) ≅ D lim Hom_{D(Mod C)}(X_{i}, Z[3])
≅ D Hom_{D(Mod C)}(hocolimX_{i}, Z[3])
≅ D Hom_{D(Mod C)}(P_{X}, Z[3]).

Note that since D^{−}(mod C) is a full subcategory of D(Mod C), we get the isomorphism

D Hom_{D^{−}(mod C)}(Z, X) ⟶^{∼} Hom_{D^{−}(mod C)}(X, Z[3]).

□

For each X ∈ E, there is a unique minimal conflation (up to isomorphism)

0 → T_{X}^{1} → T_{X}^{0} → X → 0

with T_{X}^{0}, T_{X}^{1} ∈ add T. As in [37], put

ind_{T} (X) = [T_{X}^{0}] − [T_{X}^{1}] in K_{0}(add T).

By the proof of Theorem 3.3, we have

ind_{T} (X) = ∑_{i=1}^{n}⟨F X, S_{i}⟩_{τ} [T_{i}].

The following result is easily deduced from Theorem 2.3 of [13].

Lemma 4.2. If X is a rigid object of E, then X is determined up to isomorphism by ind_{T} (X), i.e. if Y is rigid and ind_{T} (X) = ind_{T} (Y), then X is isomorphic to Y.

Proof. Since ind_{T} (X) = ind_{T} (Y), we have ind\underline{T} (X) = ind\underline{T} (Y) in the stable category \underline{E}. By Theorem 2.3 of [13], we have X ≅ Y in \underline{E}. Thus, there are E-projectives P_{X} and P_{Y} such that X ⊕ P_{X} ≅ Y ⊕ P_{Y} in E. Consider the minimal right T -approximation of X ⊕ P_{X}

0 → T^{1} → T^{0} → X ⊕ P_{X} → 0,

we have ind_{T} (X ⊕ P_{X}) = ind_{T} (Y ⊕ P_{Y}) = [T^{0}] − [T^{1}]. Note that

ind_{T} (X) = ind_{T} (X ⊕ P_{X}) − [P_{X}] = ind_{T} (Y ⊕ P_{Y}) − [P_{Y}] = ind_{T} (Y),

which implies [P_{X}] = [P_{Y}] in K_{0}(add T). Thus, we have P_{X} ≅ P_{Y} and X ≅ Y in E. □

4.2. g-vector. Let us recall the definition of g-vectors from section 7 of [21]. Let 1 < r ≤ n be integers. Let B̃ = (b̃_{ij}) be an n × r matrix with integer entries, whose principal part B (i.e. the submatrix formed by the first r rows) is antisymmetric. Let A(B̃) be the cluster algebra with coefficients associated with B̃, cf. the end of section 2.1. Let z be an element of A(B̃). Suppose that we can write z as

z = R(ŷ_{1}, …, ŷ_{r}) ∏_{i=1}^{n} x_{i}^{g_{i}}, where ŷ_{j} = ∏_{i=1}^{n} xb̃_{ij} ; i,

where R(ŷ_{1}, …, ŷ_{r}) is a primitive rational polynomial. If rank B̃ = r, then the g-vector of z is defined by

g(z) = (g_{1}, …, g_{r}).

Note that rank B̃ = r implies that the g-vector is well-defined.

As in the previous section, we let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents. We assume that E is Hom-finite and that the stable category C = \underline{E} is 2-Calabi-Yau (cf. section 2.4). Moreover, we assume that E admits a cluster-tilting object T and we write C = End_{E}(T) and \underline{C} = End_{C}(T). Let T_{1}, T_{2}, …, T_{n} be the pairwise non isomorphic indecomposable direct summands of T numbered in such a way that T_{i} is projective iff r < i ≤ n. We define B(T) = (b_{ij})_{n×n} to be the antisymmetric matrix associated with the quiver of the endomorphism algebra of T. Let B(T)^{0} be the submatrix formed by the first r columns of B(T). We suppose that we have rank B(T)^{0} = r. In analogy with the definition of g-vectors in a cluster algebra, for M ∈ E, if we can write X′_{M} as

X′_{M} = R(ŷ_{1}, …, ŷ_{r}) ∏_{i=1}^{n} x_{i}^{g_{i}}, where ŷ_{j} = ∏_{i=1}^{n} x_{i}^{b_{ij}},

where R(ŷ_{1}, …, ŷ_{r}) is a primitive rational polynomial, then we define the g-vector g_{T} (X′_{M}) of M with respect to T to be

g_{T} (X′_{M}) = (g_{1}, …, g_{r}).

As in the cluster algebra case, this is well-defined since rank B(T)^{0} = r.

Proposition 4.3. Assume that rank B(T)^{0} = r. For arbitrary M ∈ E, the g-vector g_{T} (X′_{M}) is well-defined and its i-th coordinate is given by

g_{T} (X′_{M})(i) = [ind_{T} (M) : T_{i}], 1 ≤ i ≤ r.

Proof. By the relative 3-Calabi-Yau property of D^{−}(mod C), for 1 ≤ i ≤ n, 1 ≤ j ≤ r, we have

⟨S_{i}, S_{j}⟩_{3} = [S_{i}, S_{j}] − ^{1}[S_{i}, S_{j}] + ^{2}[S_{i}, S_{j}] − ^{3}[S_{i}, S_{j}]
= [S_{i}, S_{j}] − ^{1}[S_{i}, S_{j}] + ^{1}[S_{j}, S_{i}] − [S_{j}, S_{i}]
= ^{1}[S_{j}, S_{i}] − ^{1}[S_{i}, S_{j}]
= b_{ij},

where the last equality follows from the definition of B(T). Recall the definition of X′_{M}

X′_{M} = ∏_{i=1}^{n} x_{i}^{⟨F M,S_{i}⟩_{τ}} ∑_{e} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{i=1}^{n} x_{i}^{−⟨e,S_{i}⟩_{3}}.

Let e be the dimension vector of a C-submodule of Ext_{E}^{1}(T, M) and e_{j} its j-th coordinate in the basis of the S_{i}, 1 ≤ i ≤ n. Then we have

−⟨e, S_{i}⟩_{3} = − ∑_{j=1}^{r} e_{j}⟨S_{j}, S_{i}⟩_{3} = ∑_{j=1}^{r} b_{ij}e_{j}.

Therefore, we get

∏_{i=1}^{n} x_{i}^{−⟨e,S_{i}⟩_{3}} = ∏_{i=1}^{n} x_{i}^{∑_{j=1}^{r} b_{ij}e_{j}} = ∏_{j=1}^{r} ŷ_{j} ^{e_{j}}.

Thus, we can write

X′_{M} = ∏_{i=1}^{n} x_{i}^{⟨F M,S_{i}⟩_{τ}} (∑_{e} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{j=1}^{r} ŷ_{j} ^{e_{j}}).

The polynomial

R(ŷ_{1}, …, ŷ_{r}) = ∑_{e} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{j=1}^{r} ŷ_{j} ^{e_{j}}

is primitive since it has constant term 1. Thus, by definition we have g_{T} (X′_{M})(i) = ⟨F M, S_{i}⟩_{τ} = [ind_{T} (M) : T_{i}]. □

Corollary 4.4. As above, let E be a Hom-finite k-linear Frobenius category such that its stable category C = \underline{E} is 2-Calabi-Yau and assume that

• E admits a cluster-tilting object T with indecomposable direct summands T_{1}, …, T_{n} numbered in such a way that T_{i} is projective iff r < i ≤ n, where 1 < r ≤ n is an integer;

• the first r columns of the antisymmetric matrix B(T) associated with the quiver of

the algebra C = End_{E}(T) are linearly independent. Then the following hold. a) The map M ↦ X′_{M} induces an injection from the set of isomorphism classes of

non projective rigid indecomposables of E into the set ℚ(x_{1}, …, x_{n}).

b) Let I be a finite set and T^{i}, i ∈ I, cluster tilting objects of E. Suppose that for each i ∈ I, we are given an object M_{i} which belongs to add T^{i} and does not have non zero projective direct factors. If the M_{i} are pairwise non isomorphic, then the X′_{M_{i}} are linearly independent.

Proof. a) clearly follows from b). Let us prove b). First, we will show that we can assign a degree to each x_{i} such that for every 1 ≤ i ≤ r the degree of ŷ_{i} is 1.

Indeed, it suffices to put deg(x_{i}) = k_{i}, where the k_{i} are rationals such that we have

(k_{1}, k_{2}, …, k_{n})B(T)^{0} = (1, 1, …, 1).

Since rank B(T)^{0} = r, this equation does admit a solution. Thus, the term of strictly minimal total degree in X′_{M_{j}} is

∏_{i=1}^{n} x_{i}^{[ind_{T} (M_{j}):T_{i}]}.

Suppose that the X′_{M_{i}} are linearly dependent, i.e. there is a non-empty subset I′ of I and rationals c_{i}, i ∈ I′, which are all non zero such that

∑_{i∈I′} c_{i}X′_{M_{i}} = 0.

If we consider the terms of minimal total degree of the polynomial above, we find

∑_{j∈I′′} c_{j} ∏_{i=1}^{n} x_{i}^{[ind_{T} (M_{j}):T_{i}]} = 0

for some non-empty subset I′′ of I. Since the M_{j} are all pairwise non isomorphic, Lemma 4.2 implies that the indices ind_{T} (M_{j}) are all distinct. Thus, the monomials ∏_{i=1}^{n} x_{i}^{[ind_{T} (M_{j}):T_{i}]} are linearly independent. Contradiction. □

Remark 4.5. If the algebra C has finite global dimension, then the condition rank B(T)^{0} = r is superfluous. Indeed, let A be the Cartan matrix of C. Then B(T)^{0} is the submatrix formed by the first r columns of the invertible matrix A^{−t}.

Next we will investigate the relation between the indices of an exchange pair.

Recall that F is the functor Hom_{E}(T,?) : E → mod C. A conflation of E

0 → X → Y → Z → 0

is F -exact if

0 → F X → F Y → F Z → 0

is exact in mod C. The F -exact sequences define a new exact structure on the additive category E. For each X, we have an F -exact conflation

0 → T_{1} → T_{0} → X → 0.

This shows that E endowed with the F -exact sequences has enough projectives and that its subcategory of projectives is add T. Moreover, if we denote by Ext_{F}^{i} (X, Z) the i-th extension groups of the category E endowed with the F -exact sequences, then Ext_{F}^{1} (X, Z) is the cohomology at Hom_{E}(T_{1}, Z) of the complex

0 → Hom_{E}(X, Z) → Hom_{E}(T_{0}, Z) → Hom_{E}(T_{1}, Z) → 0 → ….

Lemma 4.6. For X, Z ∈ E, there is a functorial isomorphism

Ext_{F}^{i} (X, Z) ⟶^{∼} Ext_{C}^{i}(F X, F Z).

Proof. Clearly, the derived functor

𝕃F : D^{b}(E) → D^{b}(mod C)

is fully faithful. Thus, Ext_{F}^{i} (X, Z) ⟶^{∼} Ext_{C}^{i}(F X, F Z). □

Now Proposition 15.4 of [23] still holds in our general setting.

Proposition 4.7. Let T and R be cluster tilting objects of E. Let

η′ : 0 → R_{k} → R′ → R_{k}^{∗} → 0, η′′ : 0 → R_{k}^{∗} → R′′ → R_{k} → 0

be the two exchange sequences associated to an indecomposable direct summand R_{k} of R which is not E-projective. Then exactly one of η′ and η′′ is F -exact. Moreover, we have

\underline{dim} Hom_{E}(T, R_{k}) + \underline{dim} Hom_{E}(T, R_{k}^{∗}) = max\{\underline{dim} Hom_{E}(T, R′), \underline{dim} Hom_{E}(T, R′′)\}.

Proof. Using Lemma 4.6, the proof is the same as that of proposition 15.4 in [23]. □

Corollary 4.8. Under the assumptions of the above proposition, put

I′ = ind_{T} (R′) − ind_{T} (R_{k}),
I′′ = ind_{T} (R′′) − ind_{T} (R_{k}).

Then we have

ind_{T} (R_{k}^{∗}) = \{ \begin{matrix}I′, if \underline{dim} F I′ ≥ \underline{dim} F I′′, \\ I′′, if \underline{dim} F I′ ≤ \underline{dim} F I′′. \end{matrix}

and exactly one of these cases occurs. Let h(i) = [ind_{T} (R′) − ind_{T} (R′′) : T_{i}], for 1 ≤ i ≤ n. Then h is a linear combination of the columns of B(T)^{0}.

Proof. The first part follows from Proposition 4.7 directly, because the index is additive on F -exact sequences.

Since (R_{k}, R_{k}^{∗}) is an exchange pair, we have

X′_{R_{k}}X′_{R_{k}^{∗}} = X′_{R′} + X′_{R′′}.

For simplicity, we write

H_{M} = ∑_{e} χ(Gr_{e}(Ext_{E}(T, M))) ∏_{i=1}^{r} ŷ_{i} ^{e_{i}}

for X′_{M}. By Proposition 4.3, we have

∏_{i=1}^{n} x_{i}^{[ind_{T} (R_{k})+ind_{T} (R_{k}^{∗}):T_{i}]}H_{R_{k}}H_{R_{k}^{∗}} = ∏_{i=1}^{n} x_{i}^{[ind_{T} (R′):T_{i}]}H_{R′} + ∏_{i=1}^{n} x_{i}^{[ind_{T} (R′′):T_{i}]}H_{R′′}.

Assume that ind_{T} (R_{k}^{∗}) = ind_{T} (R′) − ind_{T} (R_{k}). We have

H_{R_{k}}H_{R_{k}^{∗}} − H_{R′} = ∏_{i=1}^{n} x_{i}^{[ind_{T} (R′′)−ind_{T} (R′):T_{i}]}H_{R′′}.

By comparing the minimal total degree we get that ∏_{i=1}^{n} x_{i}^{[ind_{T} (R′′)−ind_{T} (R′):T_{i}]} is a monomial in ŷ_{i}, 1 ≤ i ≤ r, which implies the result. □

5. Frobenius 2-Calabi-Yau realizations

Recall the bijection defined in section 2.2 between antisymmetric integer n × n-matrices and finite quivers without loops nor 2-cycles with vertex set {1, 2, …, n}: The quiver Q corresponds to the matrix B iff b_{ij} > 0 exactly when there are arrows from i to j in Q and in this case their number is b_{ij}.

We call an n × n antisymmetric integer matrix B acyclic if the corresponding quiver Q does not have oriented cycles. Two matrices B and B′ are called mutation equivalent if we can obtain B′ from B by a series of matrix mutations followed by conjugation with a permutation matrix.

Let 0 ≤ r < n be positive integers and let (Q, F) be an ice quiver (cf. section 2.2) with vertex set Q_{0} = {1, …, n} and set of frozen vertices F = {r + 1, …, n}. We define B̃ to be the n × r matrix formed by the first r columns of the skew-symmetric matrix associated with Q and we let A(Q, F) = A(B̃) be the cluster algebra with coefficients associated with B̃, cf. sections 2.1 and 2.2.

Definition 5.1. A Frobenius 2-Calabi-Yau realization of the cluster algebra A(B̃) is a Frobenius category E with a cluster tilting object T as in section 3 such that

1) E has a cluster structure in the sense of [2], cf. section 2.7;

2) T has exactly n indecomposable pairwise non isomorphic summands T_{1}, T_{2}, …, T_{n} and among these, precisely T_{r+1}, …, T_{n} are projectives;

3) The matrix B̃ equals the matrix formed by the first r columns of the antisymmetric matrix associated with the quiver of the endomorphism algebra of T in E.

Remark 5.2. Suppose we have a Frobenius 2-CY realization of a cluster algebra A(Q, F) as above. Let 1 ≤ s ≤ r. Then by Lemma 2.5 b), we have conflations

0 → T_{s}^{∗} → E → T_{s} → 0,
0 → T_{s} → E′ → T_{s}^{∗} → 0.

Here the middle terms are the sums

E = ⨁_{b_{is}>0} T_{i}^{b_{is}} , E′ = ⨁_{b_{is}<0} T_{i}^{−b_{is}}.

Therefore, none of the first r vertices of Q can be a source or a sink.

Example 5.3. All quivers obtained from Theorem 2.3 of [23] and, more generally, from Theorem II.4.1 of [2] admit Frobenius 2-Calabi-Yau realizations. We illustrate this on the following specific case taken from section II.4 of [loc. cit.]. Let Δ be the graph

[figure: 2; 1; 3 .]

Let Λ be the completion of the preprojective algebra of Δ and W the Weyl group associated with Δ. Let w be the element of W given by the reduced word s_{2}s_{1}s_{2}s_{3}s_{2}. Let e_{i}, i = 1, 2, 3, be the primitive idempotents corresponding to the vertices of Δ. Let I_{i} = Λ(1 − e_{i})Λ. By Theorem II.2.8 of [2], the category Sub Λ/I_{w} formed by all Λ-submodules of finite direct sums of copies of Λ/I_{w} is a Frobenius category whose associated stable category is 2-Calabi-Yau; moreover, it contains the cluster-tilting object

T = Λ/I_{2} ⊕ Λ/I_{2}I_{1} ⊕ Λ/I_{2}I_{1}I_{2} ⊕ Λ/I_{2}I_{1}I_{2}I_{3} ⊕ Λ/I_{w}.

According to Proposition II.1.11 of [loc. cit.], in this decomposition, each direct factor differs from the preceding one by one indecomposable direct summand T_{i}, 1 ≤ i ≤ 5, and among these, exactly T_{3}, T_{4} and T_{5} are projective-injective. Moreover, by Theorem II.4.1 of [loc. cit.], the quiver of the cluster-tilting object is

\xymatrix{ & T_{1} \ar[dr] \\ T_{2} \ar@{-->}[ur] \ar[d] &  & T_{3} \ar[d] \ar[dll] \\ T_{4} \ar[rr] &  & T_{5} . \ar[ull]}

Using Theorem I.1.6 of [2], one can easily show that the category Sub Λ/I_{w} is a Frobenius 2-Calabi-Yau realization of the cluster algebra A(B̃) given by the matrix

B̃ = ( \begin{matrix}0 & −1 \\ 1 & 0 \\ −1 & 0 \\ 0 & −1 \\ 0 & 1 \end{matrix} ) .

We return to the general setup. Following [13] we define a cluster-tilting object T′ of E to be reachable from T if it is obtained from T by a finite sequence of mutations. We define an indecomposable rigid objects M to be reachable from T if it occurs as a direct factor of a cluster-tilting object reachable from T.

Theorem 5.4. Let 1 < r ≤ n be integers and A(B̃) the cluster algebras with coefficients associated with an initial n × r-matrix B̃ of maximal rank. Suppose that A(B̃) admits a Frobenius 2-CY realization E with cluster tilting object T.

a) The map M ↦ X′_{M} induces a bijection from the set of isomorphism classes of indecomposable rigid non projective objects of E reachable from T onto the set of cluster variables of A(B̃). Under this bijection, the cluster-tilting objects reachable from T correspond to the clusters of A(B̃).

b) The map M ↦ ind_{T} (M) is a bijection from the set of isomorphism classes of indecomposable rigid non projective objects of E reachable from T onto the set of g-vectors of cluster variables of A(B̃).

Proof. a) It follows from Theorem 3.3 c) that X′_{M} is a cluster variable for each indecomposable rigid M reachable from T and from the existence of a cluster structure on E that the map M ↦ X′_{M} is a surjection onto the set of cluster variables. The injectivity of the map M ↦ X′_{M} follows from Lemma 4.2 and Proposition 4.3. The second statement follows from the first one and the fact that E has a cluster structure. b) The map is injective by Lemma 4.2. It is surjective thanks to part a) and Proposition 4.3. □

Theorem 5.5. Let 1 < r ≤ n be integers and A(B̃) the cluster algebras with coefficients associated with an initial n × r-matrix B̃ of maximal rank. Suppose that A(B̃) admits a Frobenius 2-CY realization E with cluster tilting object T.

a) Conjecture 7.2 of [21] holds for A, i.e. cluster monomials are linearly independent.

b) Conjecture 7.10 of [21] holds for A, i.e.

1) Different cluster monomials have different g-vectors with respect to a given initial seed.

2) The g-vectors of the cluster variables in any given cluster form a ℤ-basis of the lattice ℤ^{r}.

c) Conjecture 7.12 of [21] holds for A, i.e. if (g_{1}, …, g_{r}) and (g′_{1}, …, g′_{r}) are the gvectors of one and the same cluster variable with respect to two clusters t and t′ related by the mutation at l, then we have

g′_{j} = \{ \begin{matrix}−g_{l} if j = l \\ g_{j} + [b_{jl}]_{+}g_{l} − b_{jl} min(g_{l}, 0) if j ≠ l \end{matrix}

where the b_{ij} are the entries of the r × r-matrix B associated with t and we write [x]_{+} for max(x, 0) for any integer x.

Proof. a) By Theorem 5.4 a), each cluster monomial m is the image X′_{M} of a rigid object M of E, where M does not have any non zero projective direct factor. Moreover, such an object M is unique up to isomorphism. Thus, given a set m_{1}, …, m_{N} of pairwise distinct cluster monomials, we obtain a set M_{1}, … M_{N} of pairwise non isomorphic rigid objects without projective direct factors such that X′_{M_{i}} = m_{i} for 1 ≤ i ≤ N. Thus, by Corollary 4.4 b), the images X′_{M_{i}} = m_{i} of the M_{i} are not only pairwise distinct but in fact linearly independent.

b) Let us prove 1). Let m and m′ be two distinct cluster monomials. We would like to compare their g-vectors with respect to a given initial cluster. By theorem 5.4 a), we may assume that this given cluster consists of the images under M ↦ X′_{M} of the indecomposable direct factors of T. Still by theorem 5.4 a), the monomials m and m′ are the images X′_{M} and X′_{M′} of two non isomorphic rigid objects M and M′ of E without non zero projective direct factors. Thus M and M′ are still non isomorphic in the stable category C = \underline{E}. But by theorem 2.3 of [13], non isomorphic rigid objects have distinct indices ind_{T} (M) and ind_{T} (M′). Therefore, they have distinct g-vectors by Proposition 4.3. Now let us prove 2). Let a cluster x′ be given. By theorem 5.4 a), the variables x′_{i} in x′ are the images under M ↦ X′_{M} of the indecomposable non projective direct summands T′_{i} of a cluster-tilting object T′ reachable from T. By Proposition 4.3, the g-vector of each x′_{i} is the index of T′_{i}. Now by Theorem 2.6 of [13], the indices of the indecomposable direct factors of a cluster-tilting object form a basis of the lattice K_{0}(add \underline{T}), where \underline{T} is the image of T in C. Thus the g-vectors of the x′_{i} form a basis of the lattice ℤ^{r}.

c) By Theorem 5.4 a), we may assume that under the maps M ↦ X′_{M}, the clusters t and t′ correspond to the cluster-tilting object T and another cluster-tilting object T′ obtained from T by mutation at the non projective indecomposable direct factor T_{l}. Moreover, the given cluster variable x corresponds to some non projective rigid indecomposable object X. By Proposition 4.3, the g-vectors of x with respect to t and t′ are given by the components of the indices ind_{T} (X) and ind_{T′}(X) in the bases formed by the ind_{T} (T_{i}), 1 ≤ i ≤ r, respectively the ind_{T′}(T′_{i}), 1 ≤ i ≤ r, where the T_{i} and the T′_{i} are the non projective indecomposable direct factors of T respectively T′. Now Theorem 3.1 of [13] tells us exactly how ind_{T} (X) and ind_{T′}(X) are related: Let

\xymatrix{T_{l} \ar[r] & E′ \ar[r] & T_{l}^{∗} \ar[r] & ΣT_{l} & and & T_{l}^{∗} \ar[r] & E \ar[r] & T_{l} \ar[r] & ΣT_{l}^{∗}}

be the exchange triangles associated with the mutation from T to T′. Let

ϕ_{+} : K_{0}(add T) → K_{0}(add T′) and ϕ_{−} : K_{0}(add T) → K_{0}(add T′)

be the linear maps which send the classes [T_{i}], i ≠ l, to themselves and send [T_{l}] to

ϕ_{+}([T_{l}]) = [E] − [T_{l}^{∗}] respectively ϕ_{−}([T_{l}]) = [E′] − [T_{l}^{∗}].

Then by Theorem 3.1 of [13], we have

ind_{T′}(X) = \{ \begin{matrix}ϕ_{+}(ind_{T} (X)) & if [ind_{T} (X) : T_{l}] ≥ 0 \\ ϕ_{−}(ind_{T} (X)) & if [ind_{T} (X) : T_{l}] ≤ 0. \end{matrix}

We leave it to the reader to check that this yields exactly the rule given in the assertion. □

Let B̃ be a 2r × r matrix whose principal (i.e. top r × r) part B_{0} is mutation equivalent to an acyclic matrix, and whose complementary (i.e. bottom) part is the r × r identity matrix. Let A(B̃) be the cluster algebra with the initial seed (x, B̃).

Theorem 5.6. With the above notation, the cluster algebra A(B̃) does not admit a Frobenius 2-CY realization.

Proof. Suppose that A(B̃) has a Frobenius 2-CY realization E. Then there is a cluster tilting object T of E with 2r indecomposable direct summands. Then we have B(T)^{0} = B̃. Since B_{0} is mutation equivalent to an acyclic matrix B_{c} by a series of mutations, we have a cluster tilting object T′ such that the quiver of the stable endomorphism algebra of T′ corresponds to B_{c}. Let A be the stable endomorphism algebra of T′. By the main theorem of [33], we have a triangle equivalence \underline{E} ≃ C_{A}, where C_{A} is the cluster category of A. Thus the cluster tilting graph of E is connected and every rigid object of E can be extended to a cluster tilting object of E.

Let F = Hom_{E}(T,?). Let S_{i}, 1 ≤ i ≤ 2r, be the simple modules of End_{E}(T). For each object M of E, we have the Laurent polynomial

X′_{M} = ∏_{i=1}^{2r} x_{i}^{⟨F M,S_{i}⟩_{τ}} ∑_{e} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{i=1}^{2r} x_{i}^{−⟨e,S_{i}⟩_{3}}.

Let

y_{j} = ∏_{i=1}^{2r} x_{i}^{b_{ij}}, 1 ≤ j ≤ r.

As in Proposition 4.3, we can rewrite X′_{M} as

X′_{M} = ∏_{i=1}^{2r} x_{i}^{⟨F M,S_{i}⟩_{τ}} (1 + ∑_{e≠0} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{i=1}^{r} y_{j}^{e_{j}}),

where e_{j} is the j-th coordinate of e in the basis of the S_{i}, 1 ≤ i ≤ 2r. If the indecomposable object M is rigid and not isomorphic to T_{i} for r < i ≤ 2r, then X′_{M} is a cluster variable of A(B̃). By the definition of the rational function F_{l,t} associated with the cluster variable x_{l,t} in [21], we have

F_{M} = X′_{M}(x_{1} = x_{2} = … = x_{r} = 1)
= ∏_{i=r+1}^{2r} x_{i}^{⟨F M,S_{i}⟩_{τ}} (1 + ∑_{e≠0} χ(Gr_{e}(Ext_{E}^{1}(T, M))) ∏_{j=r+1}^{2r} x_{j}^{e_{j−r}}).

Put

G_{M} = 1 + ∑_{e≠0} χ(Gr_{e} Ext_{E}^{1}(T, M)) ∏_{j=r+1}^{2r} x_{j}^{e_{j−r}}.

Note that G_{M} is always a polynomial of x_{i}, r + 1 ≤ i ≤ 2r, with constant term 1. By Proposition 5.2 in [21], we know that the polynomial F_{M} is not divisible by x_{i}, r + 1 ≤ i ≤ 2r. Now for i > r, we have ⟨F M, S_{i}⟩_{τ} ≥ 0 in general, which implies that ⟨F M, S_{i}⟩_{τ} = 0. In particular, ⟨F M, S_{i}⟩_{τ} = [ind_{T} (M) : T_{i}] = 0, for r + 1 ≤ i ≤ 2r. Consider M = ΣT_{1}, which is rigid and indecomposable, so X′_{M} is a cluster variable of the cluster algebra A(B̃). But in the Frobenius category E we have the conflation

0 → T_{1} → P → ΣT_{1} → 0,

where P is an injective hull of T_{1}, which implies

ind_{T} (M) = [P] − [T_{1}].

Thus there is always some r + 1 ≤ i ≤ 2r such that [ind_{T} (M) : T_{i}] ≠ 0. Contradiction. □

Remark 5.7. In the above notation, if B_{0} is acyclic, then it is easy to deduce that the cluster algebra A(B̃) does not have a Frobenius 2-CY realization. Indeed in this case, one of the first r vertices of Q which corresponds to B̃ is always a sink. This is incompatible with the existence of a Frobenius 2-CY realization by remark 5.2.

6. Triangulated 2-Calabi-Yau realizations

6.1. Definitions. Let B = (b_{ij})_{n×n} be an antisymmetric integer matrix and A(B) the associated cluster algebra. A 2-Calabi-Yau triangulated category C is called a triangulated 2-Calabi-Yau realization of the matrix B if C admits a cluster tilting object T such that

• C has a cluster structure in the sense [2], cf. section 2.4;

• T has exactly n non isomorphic indecomposable direct summands T_{1}, …, T_{n};

• The antisymmetric matrix B(T) associated with the quiver of the endomorphism

algebra of T equals B. We denote a triangulated 2-CY realization of B by C ⊃ add T.

Let n_{1} and n_{2} be positive integers. Let B_{1} and B_{2} be antisymmetric integer n_{1} × n_{1} resp. n_{2} × n_{2}-matrices. Let B_{21} be an integer n_{2} × n_{1}-matrix with non negative entries. Let C_{i} ⊃ T_{i} be a triangulated 2-CY realization of B_{i}, i = 1, 2. Let B be the matrix

( \begin{matrix}B_{1} & −B_{21}^{t} \\ B_{21} & B_{2} \end{matrix} ) .

A gluing of C_{1} ⊃ T_{1} with C_{2} ⊃ T_{2} with respect to B is a triangulated 2-CY realization C ⊃ T of B endowed with full additive subcategories T′_{1} and T′_{2} such that

• Hom_{C}(T′_{1}, T′_{2}) = 0;

• The set indec(T ) is the disjoint union of indec(T′_{1}) with indec(T′_{2});

• There is a triangle equivalence

⊥(ΣT′_{1})/(T′_{1}) ⟶^{∼} C_{2}

inducing an equivalence T′_{2} ⟶^{∼} T_{2};

• There is a triangle equivalence

⊥(ΣT′_{2})/(T′_{2}) ⟶^{∼} C_{1}

inducing an equivalence T′_{1} ⟶^{∼} T_{1}.

A principal gluing of C_{1} ⊃ T_{1} is a gluing of C_{1} ⊃ T_{1} with C_{2} ⊃ T_{2} with respect to

( \begin{matrix}B_{1} & −I_{n_{1}} \\ I_{n_{1}} & 0 \end{matrix} ) ,

where C_{2} is the cluster category of (A_{1})^{n_{1}} and T_{2} the image of the subcategory of finitely generated projective modules.

It is well known that each acyclic matrix B admits a triangulated 2-CY realization C_{Q_{B}}, where C_{Q_{B}} is the cluster category of the quiver Q_{B} corresponding to B. In the last subsection, we will see that C_{Q_{B}} does admit a principal gluing.

Conjecture 6.1. If C_{1} and C_{2} are algebraic, a gluing exists for any matrix B_{21} with non negative entries.

Amiot’s work [1] provides some evidence for the conjecture: Indeed, if C_{1} and C_{2} are generalized cluster categories [1] associated with Jacobi-finite quivers with potential [15], it is easy to construct a quiver with potential which provides a gluing as required by the conjecture.

6.2. Cluster algebras with coefficients. Let B be an antisymmetric integer n × nmatrix. Suppose that the matrix B admits a triangulated 2-CY realization C with the cluster tilting subcategory T = add T. Let T_{i}, 1 ≤ i ≤ n, be the non isomorphic indecomposable direct summands of T. By the definition, we have B(T) = B. The mutations of the matrix B correspond to the mutations of the cluster tilting object T. Fix an integer 0 < r ≤ n and consider the submatrix B^{0} of B formed by the first r columns of B. If l ≤ r, then we have

μ_{l}(B^{0}) = (μ_{l}(B))^{0},

where μ_{l} is the mutation in the direction l. Thus we can view the cluster algebra A(B^{0}) with coefficients as a sub-cluster algebra of A(B), cf. Ch. III of [2].

Denote by P the full subcategory of C whose objects are the finite direct sums of copies of T_{r+1}, …, T_{n}. We define a subcategory of C

U = ^{⊥}(ΣP) = {X ∈ C| Ext_{C}^{1}(T_{i}, X) = 0 for r < i ≤ n}.

By Theorem I.2.1 of [2], the quotient category U/P is a 2-Calabi-Yau triangulated category and the projection U → U/P induces a bijection between the cluster tilting subcategories of C containing P and the cluster tilting subcategories of U/P. Thus, a mutation of a cluster tilting object in U/P can be viewed as a mutation of a cluster tilting object in U ⊂ C which does not affect the direct summands T_{i}, r < i ≤ n. This exactly corresponds to a mutation of the matrix B in one of the first r directions. In particular, a mutation of the cluster algebra A(B^{0}) corresponds to a mutation of a cluster tilting object in U.

Recall from section 2.5 that on C, we have Palu’s cluster character associated with T, which is given by the formula

X_{M} = X_{M}^{T} = ∏_{i=1}^{n} x_{i}^{−[coind_{T} M:T_{i}]} ∑_{e} χ(Gr_{e}(Hom_{C}(T, M))) ∏_{i=1}^{n} x_{i}^{⟨S_{i},e⟩_{a}}.

We consider the composition of this map with the shift:

X′_{M} = X_{ΣM} = ∏_{i=1}^{n} x_{i}^{[ind_{T} M:T_{i}]} ∑_{e} χ(Gr_{e}(Hom_{C}(T, ΣM))) ∏_{i=1}^{n} x_{i}^{⟨S_{i},e⟩_{a}}.

We consider the restriction of the map M ↦ X′_{M} to the subcategory U. It follows from Proposition 2.3 that if M is an indecomposable rigid object reachable from T in U, then X′_{M} is a cluster variable of A(B^{0}). We will rewrite this variable so as to express its g-vector (if it is defined) in terms of the index of M: Let M be an object of U. Then Hom_{C}(T, ΣM) is an End_{C}(T)-module which vanishes at each vertex r < i ≤ n. Let e be the image of Hom_{C}(T, ΣM) in the Grothendieck group of mod End_{C}(T). Let e_{j} be the j-th coordinate of e with respect to the basis S_{i}, 1 ≤ i ≤ n. We have

⟨S_{i}, e⟩_{a} = ⟨S_{i}, e⟩_{τ} − ⟨e, S_{i}⟩_{τ}
= ∑_{j=1}^{r} e_{j}(⟨S_{i}, S_{j}⟩_{τ} − ⟨S_{j}, S_{i}⟩_{τ})

= ∑_{j=1}^{r} e_{j}(Ext_{End_{C}(T)}^{1}(S_{j}, S_{i}) − Ext_{End_{C}(T)}^{1}(S_{i}, S_{j}))

= ∑_{j=1}^{r} b_{ij}e_{j}.

As in section 4, put

y_{j} = ∏_{i=1}^{n} x_{i}^{b_{ij}} , for 1 ≤ j ≤ r.

Then X′_{M} can be rewritten as

X′_{M} = ∏_{i=1}^{n} x_{i}^{[ind_{T} (M):T_{i}]}(1 + ∑_{e≠0} χ(Gr_{e}(Hom_{C}(T, ΣM))) ∏_{j=1}^{r} y_{j}^{e_{j}}).

As in section 4, when rank B^{0} = r, we can define the g-vector of M ∈ U with respect to a cluster tilting object T. Thus we have proved part a) of the following proposition. We leave the easy proof of part b) to the reader.

Proposition 6.2. Suppose that rank B^{0} = r. Let M be an object of U.

a) The g-vector of X′_{M} with respect to the initial cluster is given by

g_{T} (X′_{M})(i) = [ind_{T} (M) : T_{i}], for 1 ≤ i ≤ r.

b) The index of the image of M in U/P with respect to the image of T is

∑_{i=1}^{r} g_{T} (X′_{M})(i)[T_{i}].

In analogy with the definition in section 5, we define a cluster tilting object T′ of U to be reachable from T if it is obtained from T by a sequence of mutations at indecomposable rigid objects of U not in P. We define an indecomposable rigid object of U to be reachable from T if it is a direct factor of a cluster tilting object reachable from T.

Theorem 6.3. Let B be an antisymmetric integer n × n-matrix and 1 ≤ r ≤ n an integer such that the submatrix B^{0} of B formed by the first r columns has rank r. Let A = A(B^{0}) be the associated cluster algebra with coefficients. Assume that the matrix B admits a triangulated 2-CY realization given by a triangulated category C with a cluster tilting object T which is the sum of n indecomposable direct factors T_{1}, …, T_{n}. Denote by P the full subcategory of C whose objects are the finite direct sums of copies of T_{r+1}, …, T_{n} and define the subcategory U of C by

U = ^{⊥}(ΣP) = {X ∈ C| Ext_{C}^{1}(T_{i}, X) = 0 for r < i ≤ n}.

For M ∈ C, define (cf. section 2.5)

X′_{M} = X_{ΣM}^{T} = ∏_{i=1}^{n} x_{i}^{[ind_{T} M:T_{i}]} ∑_{e} χ(Gr_{e}(Hom_{C}(T, ΣM))) ∏_{i=1}^{n} x_{i}^{⟨S_{i},e⟩_{a}}.

Then the following hold.

a) The map M ↦ X′_{M} induces a bijection from the set of isomorphism classes of indecomposable rigid objects of U not belonging to P and reachable from T onto the set of cluster variables of A(B^{0}). Under this bijection, the cluster-tilting objects of U reachable from T correspond to the clusters of A(B^{0}).

b) The map M ↦ [ind_{T} (M) : T_{i}]_{1≤i≤r} is a bijection from the set of indecomposable rigid objects of U not belonging to P and reachable from T onto the set of g-vectors of cluster variables of A(B^{0}).

c) Conjecture 7.2 of [21] holds for A, i.e. the cluster monomials are linearly independent over ℤ. Moreover, the cluster monomials form a basis of the ℤ[x_{r+1}, …, x_{n}]- submodule of A(B^{0}) which they generate. d) Conjecture 7.10 of [21] holds for A, i.e.

1) Different cluster monomials have different g-vectors with respect to a given initial seed.

2) The g-vectors of the cluster variables in any given cluster form a ℤ-basis of the lattice ℤ^{r}.

e) Conjecture 7.12 of [21] holds for A, i.e. if (g_{1}, …, g_{r}) and (g′_{1}, …, g′_{r}) are the gvectors of one and the same cluster variable with respect to two clusters t and t′ related by the mutation at l, then we have

g′_{j} = \{ \begin{matrix}−g_{l} if j = l \\ g_{j} + [b_{jl}]_{+}g_{l} − b_{jl} min(g_{l}, 0) if j ≠ l \end{matrix}

where the b_{ij} are the entries of the r × r-matrix B associated with t and we write [x]_{+} for max(x, 0) for any integer x.

Proof. It follows from Proposition 2.3 that the map M ↦ X′_{M} is well-defined and surjective onto the set of cluster variables of A(B^{0}). It is injective by Proposition 6.2 b) because rigid objects of U/P are determined by their indices and the map taking a rigid object M of U without non zero direct factors in P to its image in U/P is injective (up to isomorphism). This also implies part b). The same proof as for Corollary 4.4 b) yields the linear independence of the cluster monomials in c). Let us prove that the cluster monomials form a basis of the ℤ[x_{r+1}, …, x_{n}]-submodule of A(B^{0}) which they generate. Indeed, over ℤ, this submodule is spanned by the images X′_{M} of all rigid objects of U obtained as direct sums of objects of P and indecomposable rigid objects reachable from T not belonging to P. Such objects M are in particular rigid in T and they can be distinguished (up to isomorphism) by their indices. Now again, the same proof as for Corollary 4.4 b) shows that these X′_{M} are linearly independent over ℤ. Clearly this implies that the cluster monomials form a basis of the ℤ[x_{r+1}, …, x_{n}]-submodule of A(B^{0}) which they generate. As in the proof of Theorem 5.5 b), the assertions in part d) follow from the interpretation of the g-vector given in 6.2 b) and the facts that

1) rigid objects of U/P are determined by their indices (Theorem 2.3 of [13]) and 2) the indices of the indecomposable direct factors of a cluster-tilting subcategory T

of U/P form a basis of K_{0}(T ) (Theorem 2.6 of [13]).

Part e) is proved in exactly the same way as the corresponding statement for cluster algebras with a 2-CY Frobenius realization in Theorem 5.4 c). □

Example 6.4. Let A_{4} be the quiver 3 → 1 → 2 ← 4, C_{Q} the corresponding cluster category. The following is the AR quiver of C_{Q}, where P_{i}, 1 ≤ i ≤ 4, are the indecomposable projective kQ-modules.

[figure: P_{4}; 123; ΣP_{3}; …; P_{2}; 12; ΣP_{1}; …; …; P_{1}; 124; 2; ΣP_{2}; …; P_{3}; S_{1}; M; ΣP_{4}]

Let T = P_{1}⊕P_{2}⊕P_{3}⊕P_{4} be the canonical cluster tilting object in C_{Q}, P = add(P_{3}⊕P_{4}). It is easy to see that the indecomposable objects in U/P ≅ C_{A_{2}} are exactly P_{1}, P_{2}, M, ΣP_{1}, ΣP_{2}.

In this case, the matrix B(T)^{0} is

( \begin{matrix}0 & 1 \\ −1 & 0 \\ 1 & 0 \\ 0 & 1 \end{matrix} ) .

We have rank B(T)^{0} = 2. Moreover, the cluster algebra A(B(T)^{0}) has principal coefficients.

6.3. Cluster algebras with principal coefficients. In this subsection, we suppose that 2r = n and that the complementary part of B^{0} is the r × r identity matrix. Thus the cluster algebra A(B^{0}) has principal coefficients. Recall that for the matrix B, we have a triangulated 2-CY realization C ⊃ add T and we have fixed P = add(T_{r+1} ⊕ … ⊕ T_{2r}). Let Q = add(T_{1} ⊕ … ⊕ T_{r}). Let C_{1} = U/P and C_{2} = ^{⊥}(ΣQ)/Q be the quotient categories, T_{1} = add(π_{1}(T_{1} ⊕ … ⊕ T_{r})) and T_{2} = add(π_{2}(T_{r+1} ⊕ … ⊕ T_{2r})) the corresponding cluster tilting subcategories, where π_{1} and π_{2} are the respective projection functors. Then C is a gluing of C_{1} ⊃ T_{1} with C_{2} ⊃ T_{2} with respect to the matrix B.

As in section 5, for a cluster variable x_{l,t} of the cluster algebra A(B^{0}) which corresponds to an indecomposable rigid object M ∈ U and not in P, we denote the rational function F_{l,t} defined in section 3 of [21] by F_{M}. Since x_{l,t} = X′_{M}, we have

F_{M} = X′_{M}(x_{1} = … = x_{r} = 1)
= ∏_{i=r+1}^{2r} x_{i}^{[ind_{T} (M):T_{i}]}(1 + ∑_{e≠0} χ(Gr_{e}(Hom_{C}(T, ΣM))) ∏_{j=r+1}^{2r} x_{j}^{e_{j−r}}).

The following result is now a consequence of Proposition 3.6 and 5.2 in [21]. We give a proof based on representation theory. Note that conjecture 5.4 of [21] will be proved in full generality in [16].

Theorem 6.5. Conjecture 5.4 of [21] holds for A(B^{0}), i.e. the polynomial F_{M} has constant term 1. Thus we have

F_{M} = 1 + ∑_{e≠0} χ(Gr_{e}(Hom_{C}(T, ΣM))) ∏_{j=r+1}^{2r} x_{j}^{e_{j−r}}.

Proof. We need to show that for each i > r, [ind_{T} (M) : T_{i}] is zero. Since X′_{M} is a cluster variable and M is indecomposable, we have the following two cases: Case 1: M ≅ ΣT_{j} for some j ≤ r. We have ind_{T} (M) = −[T_{j}], which implies that [ind_{T} (M) : T_{i}] = 0. Case 2: M is not isomorphic to ΣT_{j} for any j ≤ r. Recall that by assumption, M is not isomorphic to T_{j} for any j > r. We have the following minimal triangle

T_{M}^{1} → T_{M}^{0} → M → ΣT_{M}^{1}

with T_{M}^{0}, T_{M}^{1} in add T and ind_{T} (M) = [T_{M}^{0}] − [T_{M}^{1}]. Since M belongs to U, for each i > r we have Hom_{C}(M, ΣT_{i}) = 0. If we had [T_{M}^{1} : T_{i}] ≠ 0 for some i > r, then the above minimal triangle would have a non zero direct factor

T_{i} → T_{i} → 0 → ΣT_{i}.

Suppose that we have [T_{M}^{0} : T_{i}] ≠ 0 for some i > r. Applying the functor F = Hom_{C}(T,?) to the triangle, we get a minimal projective resolution of F M as an End_{C}(T)-module. Note that for i > r, the projective module F T_{i} is also a simple module, which implies that F M is decomposable. Contradiction. □

Suppose that the indecomposable rigid object M of C is reachable from T and consider the polynomial F_{M} of Theorem 6.5. We define the f-vector f_{T} (M) = (f_{1}, …, f_{r}) of M with respect to T by

F_{M}|_{T rop(u_{1},...,u_{r})}(u_{1}^{−1}, …, u_{r}^{−1}) = u_{1}^{−f_{1}} … u_{r}^{−f_{r}} ,

where Trop(u_{1}, …, u_{r}) is the tropical semifield defined in section 2.1.

Proposition 6.6. Suppose that M is not isomorphic to T_{i} for 1 ≤ i ≤ 2r, and let \underline{dim} Hom_{C}(T, ΣM) = (d_{1}, …, d_{r}). Then we have

d_{i} = f_{i}, 1 ≤ i ≤ r.

Proof. By Theorem 6.5, we have

F_{M} = 1 + ∑_{e≠0} χ(Gr_{e}(Hom_{C}(T, ΣM))) ∏_{j=r+1}^{2r} x_{j}^{e_{j−r}}.

Therefore, we obtain

F_{M}|_{T rop(u_{1},...,u_{r})}(u_{1}^{−1}, …, u_{r}^{−1}) = 1 ⊕ ⨁_{e≠0} χ(Gr_{e}(Hom_{C}(T, ΣM))) ∏_{j=1}^{r} u_{j}^{−e_{j}}
= u_{1}^{−d_{1}} … u_{n}^{−d_{r}}.
□

Under the assumptions above, we have proved that the dimension vector of Hom_{C}(T, ΣM) equals the f-vector f_{T} (M). Conjecture 7.17 of [21] states that the f-vectors coincide with the denominator vectors in general. But by recent work of A. Buan, R. Marsh and I. Reiten [5], the dimension vectors do not always coincide with the denominator vectors. In fact, as shown in [5], for a quiver Q whose underlying graph is an affine Dynkin diagram, the vector dim Hom_{C_{Q}}(T, M) is different from the denominator vector of X_{M}^{T} if M = R and R is a direct factor of T, where R is a rigid regular indecomposable of maximal quasi-length. This leads to the following minimal counterexample to Conjecture 7.17 in [21]. Let us point out that the corresponding computations already appear in [12]. In subsection 5.5 below, we will show that in many cases, the f-vector is greater or equal to the denominator vector.

6.4. A counterexample.

Example 6.7. Let Q be the following quiver

\xymatrix{ & 3 \ar[dl] \\ 1 \ar@{=>}[rr] &  & 2 . \ar@{-->}[ul]}

Let A(Q) be the cluster algebra associated with the initial seed given by Q and x = (x_{1}, x_{2}, x_{3}). Consider the mutations at 3, 2, 1. Let x^{t_{3}} be the corresponding cluster. We have

x_{1}^{t_{3}} = \frac{x_{1}^{2} + 2x_{1}x_{2} + x_{2}^{2} + x_{3}}{x_{1}x_{2}x_{3}}

and the corresponding F -polynomial is

F_{x_{1}^{t_{3}}} = 1 + (1 + y_{1} + y_{1}y_{2})y_{3} + y_{1}y_{2}y_{3}^{2}.

Then the f-vector of x_{1}^{t_{3}} does not coincide with the denominator vector.

Let us interpret this counterexample in terms of representation theory. Let A_{2,1} be the quiver

\xymatrix{ & 3 \ar[dr] \\ 1 \ar@{-->}[ur] \ar[rr] &  & 2 .}

Consider the cluster category C_{A_{2,1}} of kA_{2,1}. Let P_{i}, 1 ≤ i ≤ 3, be the indecomposable projective modules and S_{i} the corresponding simple modules. Then

T = P_{1} ⊕ P_{2} ⊕ τS_{3},

is a cluster tilting object of C_{A_{2,1}}, where τ is the Auslander-Reiten translation functor. The quiver Q_{T} of T looks like

\xymatrix{ & τS_{3} \ar@{-->}[dl] \\ P_{1} \ar@{=>}[rr] &  & P_{2} . \ar@{-->}[ul]}

We will show that the cluster category C_{A_{2,1}} ⊃ add T admits a principal gluing. For this, consider the following quiver Q_{1} :

\xymatrix{ & 6 \ar[r] & 3 \ar[d] \\ 4 \ar[r] & 1 \ar[r] \ar[u] & 2 & 5 . \ar[l]}

It admits a cluster category C_{Q_{1}}. Let T_{Q_{1}} = kQ_{1} be the canonical cluster tilting object in C_{Q_{1}}. Let T′ = μ_{3}(μ_{6}(T_{Q_{1}})) be the cluster tilting object obtained by mutations from T_{Q_{1}}. Denote the non isomorphic indecomposable direct summands of T′ by T′_{i}, 1 ≤ i ≤ 6. Then the quiver of Q_{T′} is

\xymatrix{ &  & T′_{6} \ar[d] \\  &  & T′_{3} \ar@{-->}[dl] \\ T′_{4} \ar[r] & T′_{1} \ar@{=>}[rr] &  & T′_{2} \ar@{-->}[ul] & T′_{5} . \ar[l]}

Let P = add(T′_{4} ⊕ T′_{5} ⊕ T′_{6}). Then U/P is a 2-Calabi-Yau triangulated category and admits a cluster tilting object with the quiver Q_{T}. By the main theorem of [33], we know that there is a triangle equivalence U/P ≃ C_{A_{2,1}}. Thus, we see that the matrix B(T′) admits a triangulated 2-CY realization C_{Q_{1}} which is the required principal gluing of C_{A_{2,1}} ⊃ add T. We may assume that the images of T′_{1}, T′_{2}, T′_{3} coincide with P_{1}, P_{2}, τS_{3} in C_{A_{2,1}} respectively. Denote the shift functor in C_{Q_{1}} (resp. C_{A_{2,1}}) by Σ (resp. [1]).

Let N be the preimage of S_{3} in C_{Q_{1}}. Then one can easily compute

\underline{dim} Hom_{C_{Q_{1}}} (T′, ΣN) = \underline{dim} Hom_{C_{A_{2,1}}} (T, τS_{3}) = (1, 1, 2).

Note that the denominator vector of X′_{N} equals the denominator vector of X_{τS_{3}}^{T}. Now the result follows from the Proposition above.

6.5. An inequality. Let T be a 2-Calabi-Yau triangulated category with cluster-tilting object T. Recall that we have the generalized Caldero-Chapoton map

X_{M}^{T} = ∏_{i=1}^{n} x_{i}^{−[coind_{T} (M):T_{i}]} ∑_{e} χ(Gr_{e}(GM)) ∏_{i=1}^{n} x_{i}^{⟨S_{i},e⟩_{a}},

where G is the functor Hom_{C}(T,?) : C → mod End_{C}(T). The following proposition is proved in greater generality in [16].

Proposition 6.8. For each M in T , let \underline{dim} GM = (m_{1}, …, m_{n}) and let 1 ≤ i ≤ n. We have

−[coind_{T} (M) : T_{i}] + ⟨S_{i}, e⟩_{a} ≥ −m_{i},

for each submodule N of GM with \underline{dim} N = e. Thus the exponent of x_{i} in the denominator of X_{M} is less or equal to m_{i}.

Proof. This result holds for the case M ≅ ΣT′, T′ ∈ add T obviously. We assume that M is indecomposable and not isomorphic to any ΣT′. The case where M is decomposable is a consequence of the multiplication theorem for X_{?}. Now by Lemma 7 of [37], we have

−[coind_{T} (M) : T_{i}] = −⟨S_{i}, GM⟩_{τ} .

Note that we have the short exact sequence of End_{C}(T)-modules

0 → N → GM → GM/N → 0.

By applying the functor Hom(S_{i},?), we get

⟨S_{i}, N⟩_{τ} + ⟨S_{i}, GM/N⟩_{τ} − ⟨S_{i}, GM⟩_{τ} + dim Ext^{2}(S_{i}, N) ≥ 0.

By the stable 3-Calabi-Yau property of mod End_{C}(T) proved in [34], we have dim Ext^{2}(S_{i}, N) ≤ dim Ext^{1}(N, S_{i}). Therefore, we have

−[coind_{T} (M) : T_{i}] + ⟨S_{i}, e⟩_{a} ≥ −⟨N, S_{i}⟩_{τ} − ⟨S_{i}, GM/N⟩_{τ} − dim Ext^{2}(S_{i}, N)
≥ −[N, S_{i}] − [S_{i}, GM/N] + ^{1}[S_{i}, GM/N]
≥ −m_{i}.
□

6.6. Behaviour of the g-vectors under mutation. Let B = (b_{ij}) be an antisymmetric integer r ×r-matrix. Let C ⊃ add T be a triangulated 2-CY realization of B. Let T_{1}, …, T_{r} be the non isomorphic indecomposable factors of T. Let 1 ≤ l ≤ r be an integer and T′ = μ_{l}(T) the mutation of T at T_{l}. Thus, the non isomorphic indecomposable factors of T′ are T_{1}, …, T_{l}^{∗}, …, T_{r}. Let C_{1} be a principal gluing of C ⊃ add T and C_{2} a principal gluing of C ⊃ add T′ (we assume such gluings exist). For each indecomposable object M ∈ C reachable from T, we denote by F_{M}^{T} and F_{M}^{T′} the F-polynomials of M with respect to C_{1} and C_{2} respectively. Following [21], we define the integers h_{l} and h′_{l} by

u^{h_{l}} = F_{M}^{T}|_{T rop(u)}(u^{[−b_{k1}]_{+}}, …, u^{−1}, …, u^{[−b_{kn}]_{+}}),
u^{h′_{l}} = F_{M}^{T′}|_{T rop(u)}(u^{[b_{k1}]_{+}}, …, u^{−1}, …, u^{[b_{kn}]_{+}}),

where u^{−1} is in the l-th position.

The following proposition shows that if the gluings C_{1} and C_{2} exist (for example if C is algebraic and Conjecture 6.1 holds), then Conjecture 6.10 of [21] holds for the cluster algebra with principal coefficients associated with B.

Proposition 6.9. In the above notation, we have

h′_{l} = −[[ind_{T} (M) : T_{l}]]_{+}, h_{l} = min(0, [ind_{T} (M) : T_{l}]).

Proof. Let S_{i}, 1 ≤ i ≤ r, be the top of the indecomposable right projective End_{C}(T′)- module Hom_{C}(T′, T′_{i}). First we will show that g_{l} = [ind_{T} (M) : T_{l}] > 0 iff S_{l} occurs as a submodule of the module Hom_{C}(T′, ΣM) and that the multiplicity of S_{l} in the socle of Hom_{C}(T′, ΣM) equals [ind_{T} (M) : T_{l}].

Suppose that g_{l} > 0. Then we have the following triangle

T_{M}^{1} → T_{M}^{0′} ⊕ (T_{l})^{g_{l}} → M → ΣT_{M}^{1}

with T_{M}^{1}, T_{M}^{0′} in add T and [T_{M}^{0′} : T_{l}] = 0, where (T_{l})^{g_{l}} is the sum of g_{l} copies of T_{l}. Applying the functor Hom_{C}(T′,?) to the shift of the above triangle, we get the exact sequence

0 → Hom_{C}(T′, Σ(T_{l})^{g_{l}}) → Hom_{C}(T′, ΣM) → Hom_{C}(T′, Σ^{2}T_{M}^{1}) → …

Note that Hom_{C}(T′, Σ(T_{l})^{g}) ≅ (S_{l})^{g_{l}}, i.e. S_{l} occurs with multiplicity ≥ g_{l} in the socle of Hom_{C}(T′, ΣM). If the multiplicity of S_{l} in the socle of Hom_{C}(T′, ΣM) was > g_{l}, then S_{l} would occur in the socle of Hom_{C}(T′, Σ^{2}T_{M}^{1}). This is not the case since Hom_{C}(T′, Σ^{2}T_{M}^{1}) is the sum of injective indecomposables not isomorphic to the injective hull Hom_{C}(T′, Σ^{2}T_{l}) of S_{l}. Conversely, if S_{l} occurs in the socle of Hom_{C}(T′, ΣM), thanks to the split idempotents property of C, we have an irreducible morphism α : ΣT_{l} → ΣM in C. Thus, by the definition of the index, we get g_{l} > 0. Moreover, the multiplicity of S_{l} equals g_{l} by the same argument as before.

Assume that g_{l} > 0. For an arbitrary submodule U of Hom_{C}(T′, ΣM), let dim U = (e_{1}, …, e_{n}). We will show that

e_{l} ≤ g_{l} + ∑_{i}[b_{il}]_{+}e_{i}.

Indeed, consider the projective resolution of the simple module S_{l}

… → ⊕P_{i}^{b_{il}} → P_{l} → S_{l} → 0.

Applying the functor Hom_{End_{C}(T′)}(?, U), we get the exact sequence

0 → Hom(S_{l}, U) → Hom(P_{l}, U) → Hom(⊕P_{i}^{b_{il}}, U) → …,

which implies the inequality because the dimension of Hom(S_{l}, U) is less or equal to the multiplicity of S_{l} in the socle of Hom_{C}(T′, ΣM), which equals g_{l}. By Theorem 6.5, we have

u^{h′_{l}} = F_{M}^{T′}|_{T rop(u)}(u^{[b_{k1}]_{+}}, …, u^{−1}, …, u^{[b_{kn}]_{+}})
= 1 ⊕ ⨁_{e} χ(Gr_{e}(Hom_{C}(T′, ΣM)))u^{−e_{l}} ∏_{i≠l}(u^{[b_{ki}]_{+}})^{e_{i}}.

We have just shown that for each e, we have

−e_{l} + ∑_{i}[b_{il}]_{+}e_{i} ≥ g_{l},

and the equality occurs if e is the dimension vector of the submodule (S_{l})^{g_{l}}. We conclude that we have h′_{l} = −[ind_{T} (M) : T_{l}]. If g_{l} ≤ 0, then S_{l} does not occur in the socle of Hom_{C}(T′, ΣM) and it is easy to see that h′_{l} = 0. Dually, we have the equality h_{l} = min(0, [ind_{T} (M) : T_{l}]). □

6.7. Acyclic cluster algebras with principal coefficients. Let B be an antisymmetric integer r × r-matrix. Assume that B is acyclic. Let Q be the corresponding quiver of B with set of vertices Q_{0} = {1, …, r} and with set of arrows Q_{1}. Let C_{Q} be the cluster category of Q, T = kQ the canonical cluster tilting object of C_{Q}. We claim that the cluster category C_{Q} ⊃ add T admits a principal gluing.

Indeed, we define a new quiver Q̃ = Q\overleftarrow{∐}Q_{0} associated with Q: Its set of vertices is {1, …, 2r}, and its arrows are those of Q and new arrows from r + i to i for each vertex i of Q. Since Q is acyclic, so is Q̃, hence kQ̃ is finite-dimensional and hereditary. Thus, we have the cluster category CQ̃ which is a triangulated 2-CY realization of the matrix

( \begin{matrix}B & −I_{r} \\ I_{r} & 0 \end{matrix} ) .

In particular, CQ̃ ⊃ add kQ̃ is a principal gluing for C_{Q} ⊃ add T. Thus, Proposition 6.2, Theorem 6.3, Theorem 6.5 and Proposition 6.6 hold for acyclic cluster algebras with principal coefficients.

Let P_{i}, 1 ≤ i ≤ 2r, be the non isomorphic indecomposable projective right modules of kQ̃. Let P = add(P_{r+1} ⊕ … ⊕ P_{2r}). We have a triangle equivalence

⊥(ΣP)/P ⟶^{∼} C_{Q}.

Recall that there is a partial order on ℤ^{r} defined by

α ≤ β iff α(i) ≤ β(i), for 1 ≤ i ≤ r, where α, β ∈ ℤ^{r}.

Proposition 6.10. Let B be a 2r×r integer matrix, whose principal part is antisymmetric and acyclic and whose complementary part is the identity matrix. Let σ be a sequence k_{1}, …, k_{m} with 1 ≤ k_{i} ≤ r. Denote by B_{σ} the matrix

μ_{k_{1}} ∘ μ_{k_{2}} … ∘ μ_{k_{m}}(B) = (b_{ij}^{σ}).

Let E_{σ} = (e_{1}, e_{2}, …, e_{r}) be the complementary part of B_{σ}, where e_{i} ∈ ℤ^{r}, 1 ≤ i ≤ r. Then for each i, we have e_{i} ≤ 0 or e_{i} ≥ 0.

Proof. Suppose that there is some k such that e_{k} ≰ 0 and e_{k} ≱ 0. For simplicity, assume that k = 1, i.e. there are r < i, j ≤ 2r such that b_{i1}^{σ} > 0 and b_{j1}^{σ} < 0.

Let Q be the quiver corresponding to the principal part of B and Q̃ as constructed above. By the argument above, there is a cluster tilting object T′ of C_{kQ̃} such that B(T′)^{0} = B_{σ}. We have arrows P_{i} → T′_{1} and T′_{1} → P_{j}, where T′_{1} is the indecomposable direct summand of T′ corresponding to the first column of B_{σ}. Now if we consider the mutation in direction 1 of T′, we will have an arrow P_{i} → P_{j} in Q_{μ_{1}(T′)}. But this is impossible, since for r < l ≤ 2r, the P_{l} are simple pairwise non isomorphic modules so we have

Hom_{C_{kQ̃}}(P_{i}, P_{j}) = Hom_{kQ̃}(P_{i}, P_{j}) = 0.
□

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Department of Mathematics, Sichuan University, 610064 Chengdu, P.R.China

E-mail address: flyinudream@yahoo.com.cn

U.F.R. de Mathématiques, Institut de Mathématiques, U.M.R. 7586 du CNRS, Université Denis Diderot - Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France

E-mail address: keller@math.jussieu.fr www.math.jussieu.fr/  keller